Invent. math.

DOI 10.1007/s00222-015-0583-y

Sato-Tate theorem for families and low-lying zeros of automorphic L-functions

With appendices by Robert Kottwitz [A] and by Raf Cluckers, Julia Gordon, and Immanuel Halupczok [B]

Sug Woo Shin • Nicolas Templier

Received: 17 September 2012 / Accepted: 5 January 2015

© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let G be a reductive group over a number field F which admits discrete series representations at infinity. Let LG = G xi Gal(F/F) be the associated L-group and r: LG ^ GL (d, C) a continuous homomorphism which is irreducible and does not factor through Gal(F/ F). The families under consideration consist of discrete automorphic representations of G (AF) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321-331, 1987) and Serre (J Am Math Soc 10(1):75-102, 1997). As an application we study the distribution of the low-lying zeros of the associated family of L-functions L (s,n,r), assuming from the principle of functoriality that these L-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of

S. W. Shin

Department of Mathematics, University of California Berkeley, Evans Hall, Berkeley, CA 94720, USA

S. W. Shin

School of Mathematics, KIAS, Seoul 130-722, Republic of Korea e-mail: prmideal@gmail.com

N. Templier (B)

Department of Mathematics, Cornell University, Malott Hall, Ithaca,

NY 14853-4201, USA

e-mail: templier@math.cornell.edu

Published online: 13 March 2015

Springer

the 1 -level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius-Schur indicator to determine this symmetry type. If r is not isomorphic to its dual r v then the symmetry type is unitary. Otherwise there is a bilinear form on Cd which realizes the isomorphism between r and rv. If the bilinear form is symmetric (resp. alternating) then r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

Mathematics Subject Classification 11F55 ■ 11F67 ■ 11F70 ■ 11F72 ■ 11F75 ■ 14L15 ■ 20G30 ■ 22E30 ■ 22E35

Contents

1 Introduction.........................................

2 Satake transforms......................................

3 Plancherel measure on the unramified spectrum......................

4 Automorphic L-functions..................................

5 Sato-Tate equidistribution .................................

6 Background materials....................................

7 A uniform bound on orbital integrals............................

8 Lemmas on conjugacy classes and level subgroups ....................

9 Automorphic Plancherel density theorem with error bounds................

10 Langlands functoriality...................................

11 Statistics of low-lying zeros.................................

12 Proof of Theorem 11.5...................................

Appendix A: By Robert Kottwitz................................

Appendix B: By Raf Cluckers, Julia Gordon and Immanuel Halupczok............

References............................................

1 Introduction

The non-trivial zeros of automorphic L-functions are of central significance in modern number theory. Problems on individual zeros, such as the Riemann hypothesis (GRH), are elusive. There is however a theory of the statistical distribution of zeros in families. The subject has a long and rich history. A unifying modern viewpoint is that of a comparison with a suitably chosen model of random matrices: the Katz-Sarnak heuristics. There are both theoretical and numerical evidences for this comparison. Comprehensive results in the function field case [59] have suggested an analogous picture in the number field case as explained in [60]. In a large number of cases, and with high accuracy, the distribution of zeros of automorphic L-functions coincide with the distribution of eigenvalues of random matrices. See [37,85] for numerical investigations and conjectures and see [40,49,50,53,68,82,84] and the references therein for theoretical results.

The concept of families is central to modern investigations in number theory. We want to study in the present paper certain families of automorphic representations over number fields in a very general context. The families under consideration are obtained from the discrete spectrum by imposing constraints on the local components at archimedean and non-archimedean places and by applying Langlands global functoriality principle.

Our main result is a Sato-Tate equidistribution theorem for these families (Theorem 1.3). As an application of this main result we can give some evidence towards the Katz-Sarnak heuristics [60] in general and establish a criterion for the random matrix model attached to families, i.e. for the symmetry type.

1.1 Sato-Tate theorem for families

The original Sato-Tate conjecture is about an elliptic curve E, assumed to be defined over Q for simplicity. The number of points in E (Fp) for almost all primes p (with good reduction) gives rise to an angle 6p between — n and n. The conjecture, proved in [7], asserts that if E does not admit complex multiplication then [6p} are equidistributed according to the measure 2 sin2 6d6. In the context of motives a generalization of the Sato-Tate conjecture was formulated by Serre [96].

To speak of the automorphic version of the Sato-Tate conjecture, let G be a connected split reductive group over Q with trivial center and n an automorphic representation of G(A). Here G is assumed to be split for simplicity (however we stress that our results are valid without even assuming that G is quasi-split; see Sect. 5 below for details). The triviality of center is not serious as it essentially amounts to fixing central character. Let T be a maximal split torus of G. Denote by T its dual torus and Q the Weyl group. As n = ®'vnv is unramified at almost all places p, the Satake isomorphism identifies np with a point on T/Q. The automorphic Sato-Tate conjecture should be a prediction about the equidistribution of np on T/Q with respect to a natural measure (supported on a compact subset of T/ Q). It seems nontrivial to specify this measure in general. The authors do not know how to do it without invoking the (conjectural) global L-parameter for n. The automorphic Sato-Tate conjecture is known in the limited cases of (the restriction of scalars of) GL1 and GL2 [6,7]. In an ideal world the conjecture should be closely related to Langlands functoriality.

In this paper we consider the Sato-Tate conjecture for a family of automor-phic representations, which is easier to state and prove but still very illuminating. Our working definition of a family {Fk }k^1 is that each Fk consists of all automorphic representations n of G (A) of level Nk with n^ cohomological of weight ¡k, where Nk e and is an irreducible algebraic representation of G, such that either

(1) (level aspect) is fixed, and Nk as k or

(2) (weight aspect) Nk is fixed, and m(%k) as k ^ro,

where m (%k) e R^0 should be thought of as the minimal distance of the highest weight of to root hyperplanes. (See Sect. 6.4 below for the precise definition.) Note that each Fk has finite cardinality and \Fk as k (For a technical reason Fk is actually allowed to be a multi-set. Namely the same representation can appear multiple times, for instance more than its automor-phic multiplicity.) In principle we could let and Nk vary simultaneously but decided not to do so in the current paper in favor of transparency of arguments. For instance families of type (i) and (ii) require somewhat different ingredients of proof in establishing the Sato-Tate theorem for families, and the argument would be easier to understand if we separate them. It should be possible to treat the mixed case (where both Nk and vary) by combining techniques in the two cases (i) and (ii).

Let Tc be the maximal compact subtorus of the complex torus T. The quotient Tc/ i2 is equipped with a measure TST, to be called the Sato-Tate measure, coming from the Haar measure on a maximal compact subgroup of G (of which TTc is a maximal torus). The following is a rough version of our result on the Sato-Tate conjecture for a family.

Theorem 1.1 Suppose that G (R) has discrete series representations. Let {Fk }k^1 be a family in the level aspect (resp. weight aspect) as above. Let {pk} be a strictly increasing sequence of primes such that Nk (resp. ) grows faster

than any polynomial in pk in the sense that p ^ 0 (resp. —^ 0)

as k ^ <. Assume that the members of Fk are unramified at pk for every k. Then the Satake parameters {npk: n e Fk}k^1 are equidistributed with respect to /xS1.

To put things in perspective, we observe that there are three kinds of statistics about the Satake parameters of {npk: n e Fk}k^1 depending on how the arguments vary.

(i) Sato-Tate: Fk is fixed (and a singleton) and pk ^ <.

(ii) Sato-Tate for a family: \Fk and pk ^ <.

(iii) Plancherel: \Fk \ ^ < and pk is a fixed prime.

The Sato-Tate conjecture in its original form is about equidistribution in case (i) whereas our Theorem 1.1 is concerned with case (ii). The last item is marked as Plancherel since the Satake parameters are expected to be equidistributed with respect to the Plancherel measure (again supported on Tc/ in case (iii). This has been shown to be true under the assumption that G (R) admits discrete series in [99]. We derive Theorem 1.1 from an error estimate (depending on k) on the difference between the Plancherel distribution at p and the actual

distribution of the Satake parameters at pk in Fk. This estimate (see Theorem 1.3 below) refines the main result of [99] and is far more difficult to prove in that several nontrivial bounds in harmonic analysis on reductive groups need to be justified.

1.2 Families of L-functions

An application of Theorem 1.1 is to families of L-functions. We are able to verify to some extent the heuristics of Katz and Sarnak [60] and determine the symmetry type, see Sect. 1.3 below. In this subsection we define the relevant families of L-functions and record some of their properties.

Let r: lG ^ GL(d, C) be a continuous L-homomorphism. We assume the Langlands functoriality principle: for all n e Fk there exists an isobaric automorphic representation n = r*n of GL(d, A) which is the functorial lift of the automorphic representation n of G (A), see Sect. 4.3 for a review of the concept of isobaric representations and Sect. 10 for the precise statement of the hypothesis. This hypothesis is only used in Theorem 1.5, Sects. 11 and 12. By the strong multiplicity one theorem n is uniquely determined by all but finitely many of its local factors nv = r*nv.

To an automorphic representation n on GL(d, A) we associate its principal L-function L(s, n). By definition L(s, n, r) = L(s, n). By the theory of Rankin-Selberg integrals or by the integral representations of Godement-Jacquet, L(s, n) has good analytic properties: analytic continuation, functional equation, growth in vertical strips. In particular we know the existence and some properties of its non-trivial zeros, such as the Weyl's law (Sect. 4.4).

We denote by Fk = r*Fk the set of all such n = r*n for n e Fk. Since the strong multiplicity one theorem implies that n is uniquely determined by its L-function L(s, n). We simply refer to F = r*F as a family of L-functions.

In general there are many ways to construct interesting families of L-functions. In a recent manuscript [87], Sarnak attempts to sort out these constructions into a comprehensive framework and proposes1 a working definition (see also [67]). The families of L-functions under consideration in the present paper fit well into that framework. Indeed they are harmonic families in the sense that their construction involves inputs from local and global harmonic analysis. Other types of families include geometric families constructed as Hasse-Weil L-functions of arithmetic varieties and Galois families associated to families of Galois representations.

1 Sarnak and the authors gave a more refined and updated framework in [89] while our paper was under review.

1.3 Criterion for the symmetry type

Katz and Sarnak [60] predict that one can associate a symmetry type to a family of L-functions. By definition the symmetry type is the random matrix model which is conjectured to govern the distribution of the zeros. There is a long and rich history for the introduction of this concept.

Hilbert and Polya suggested that there might be a spectral interpretation of the zeros of the Riemann zeta function. Nowadays strong evidence for the spectral nature of the zeros of L-functions comes from the function field case: zeros are eigenvalues of the Frobenius acting on cohomology. This is exemplified by the equidistribution theorem of Deligne and the results of Katz and Sarnak [59] on the distribution of the low-lying eigenvalues in geometric families.

In the number field case the first major result towards a spectral interpretation is the pair correlation of high zeros of the Riemann zeta function by Montgomery. Developments then include Odlyzko's extensive numerical study and the determination of the n-level correlation by Hejhal and Rudnick and Sarnak [86]. The number field analogue of the Frobenius eigenvalue statistics of [59] concerns the statistics of low-lying zeros.

More precisely [60] predicts that the low-lying zeros of families of L-functions are distributed according to a determinantal point process associated to a random matrix ensemble. This will be explained in more details in Sects. 1.5 and 1.6 below. We shall distinguish between the three determinantal point processes associated to the unitary, symplectic and orthogonal ensembles.2 Accordingly the symmetry type associated to a family F is defined to be unitary, symplectic or orthogonal (see Sect. 1.6 for typical results).

Before entering into the details of this theory in Sect. 1.5 below, we state here our criterion for the symmetry type of the harmonic families rxF defined above. We recall in Sect. 6.8 the definition of the Frobenius-Schur indicator s(r) e {-1, 0, 1} associated to an irreducible representation. We shall prove that the symmetry type is determined by s (r). This is summarized in the following which may be viewed as a refinement of the Katz-Sarnak heuristics.

Criterion 1.2 Let r: LG ^ GL(d, C) be a continuous L-homomorphism which is irreducible and non-trivial when restricted to G. Consider the family r*F of automorphic L-functions of degree d as above.

(i) Ifr is not isomorphic to its dual rv then s (r) = 0 and the symmetry type is unitary.

(ii) Otherwise there is a bilinear form on Cd which realizes the isomorphism between r and rv. By Schur lemma it is unique up to scalar and is either symmetric or alternating. If it is symmetric then r is real, s (r) = 1 and

2 In this paper we do not distinguish in the orthogonal ensemble between the O, SO(odd) and SO(even) symmetries. We will return to this question in a subsequent work.

the symmetry type is symplectic. If it is alternating then r is quaternionic, s (r) = —1 and the symmetry type is orthogonal.

We note that the conditions that r be irreducible and non-trivial when restricted to G are optimal. If r were trivial when restricted to G then L (s, n, r) would be constant and equal to a single Artin L-function and the low-lying zeros would correspond to the eventual vanishing of this Artin L-function at the central point (which is a different problem). Also the universality exhibited in our criterion may be compared with the GUE universality of the high zeros of [86].

If r were reducible then the L-functions would factor as a product L(s,n,r1)L(s,n,r2). Suppose that both r1 and r2 are irreducible and nontrivial when restricted to G. If r1 = r2 then clearly the distribution of zeros will be as before but with multiplicity two. If r1 ^ r2 then we expect that the zeros will follow the distribution of the independent superposition of the two random matrix ensembles attached to r1 and r2. In other words the zeros of L(s, n, r1) are uncorrelated to the zeros of L(s, n, r2), and one could verify this using the methods of this paper to some extent. In particular we expect no repulsion between the respective sequences of zeros.

It would be interesting to study families of automorphic representations over a function field k = Fq (X) of a curve X. To our knowledge the Katz-Sarnak heuristics for such families are not treated in the literature, except in the case of G = GL(1) where harmonic families coincide with the geometric families treated by Katz-Sarnak (e.g. Dirichlet L-series with quadratic character are the geometric families of hyperelliptic curves in [59, §10]). Over function fields our criterion has the following interpretation. We consider families of automorphic representations n of G (Ak); for simplicity we suppose that each automorphic representations n of G (Ak) in the family F is attached to an irreducible l-adic representation p: Gal(ksep/k) ^ LG. Then r*n is attached to the Galois representation r o p, and corresponds to a constructible l-adic sheaf F of dimension d on the curve X. The zeros of the L-function L (s, n, r) are the eigenvalues of Frobenius on the first cohomology, more precisely the numerator of the L-function L(s, n, r) is

det(1 — q—sFr|H 1(X, F)).

If s (r) = —1 [resp. s (r) = 1] then there is an alternating (resp. symmetric) pairing on the sheaf F. The natural pairing on H1 ( X, F) induced by the cup product is symmetric (resp. alternating) and invariant by the action of Frobenius. Thus the zeros of L(s, n, r) are the eigenvalues of an orthogonal (resp. symplectic) matrix. This is in agreement with the assertion (ii) of our Criterion 1.2. We also note the related situation [58].

Known analogies between L-functions and their symmetries over number fields and function fields are discussed in [60, §4]. Overall we would like

propose Criterion 1.2 and its analogue for geometric families as an answer to the question mark in the entry 6-A of Table 2 in [60].

1.4 Automorphic Plancherel density theorem with error bounds

We explain a more precise version of the theorem and method of proof for the Sato-Tate theorem for families (Sect. 1.1). The key is to bound the error terms when we approximate the distribution of local components of automorphic representations in a family with the Plancherel measure.

For simplicity of exposition let us assume that G is a split reductive group over Q with trivial center as in Sect. 1.1. A crucial hypothesis is that G (R) admits an R-anisotropic maximal torus [in which case G (R) admits discrete series representations]. Let ^disc(G) denote the set of isomorphism classes of discrete automorphic representations of G(A). We say that n e ^disc(G) has level N and weight % if n has a nonzero fixed vector under the adelic version of the full level N congruence subgroup K (N) c G (A<) and if ® % has nonzero Lie algebra cohomology. In this subsection we make a further simplifying hypothesis that % has regular highest weight, in which case as above must be a discrete series representation. (In the main body of this paper, the latter assumption on % is necessary only for the results in Sects. 9.6-9.8, where more general test functions are considered)

Define F = F(N, %) to be the finite multi-set consisting of n e ^disc(G) of level N and weight %, where each such n appears in F with multiplicity

aF(n) := dim(n<)K(N) e Z^.

This quantity naturally occurs as the dimension of the n -isotypical subspace in the cohomology of the locally symmetric space for G of level N with coefficient defined by %. The main motivation for allowing n to appear aF (n ) times is to enable us to compute the counting measure below with the trace formula.

Let p be a prime number. Write G (Qp )A for the unitary dual of irreducible smooth representations of G (Qp). The unramified (resp. unramified and tempered) part of G(Qp)A is denoted G(Qp)A'ur [resp. G(Qp)Aur-temp]. There is a canonical isomorphism

G(Qp)A'ur'temp - Tc/Q. (1.1)

The unramified Hecke algebra of G(Qp) will be denoted Hur(G(Qp)). There is a map from Hur (G(Qp)) to the space of continuous functions on Tc/ Q:

0 ^ Tdetermined by T(n) = tr n(^), Vn e G(Qp)A<ur'temp.

There are two natural measures supported on Tc/ Q. The Plancherel measure Tp1'ur, dependent on p, is defined on G(Qp)A,ur and naturally arises in local

harmonic analysis. The Sato-Tate measure /¿ST on Tc/ Q is independent of p and may be extended to G (Qp )A,ur by zero. Both /¿p1,ur and /¿ST assign volume 1 to Tc/Q. There is yet another measure ¿FT* on G(Qp)A,ur, which is the averaged counting measure for the p-components of members of F. Namely

¿F? :=F X **p (1.2)

where S„p denotes the Dirac delta measure supported at np. [Each n e ^disc(G) contributes aF(n) times to the above sum.] Our primary goal is to bound the difference between /¿p1,ur and ¿F^1. [Note that our definition of ¿¿fp in the main body will be a little different from (1.2) but asymptotically the same, see Remark 9.9.]

In order to quantify error bounds, we introduce a filtration {Hur (G(Qp))^k}KeZ>0 on Hur(G(Qp)) as a complex vector space. The filtration is increasing, exhaustive and depends on a non-canonical choice. Roughly speaking, Hur(G(Qp))^k is like the span of all monomials of degree ^ k when Hur(G(Qp)) is identified with (a subalgebra of) a polynomial algebra. For each %, it is possible to assign a positive integer m (%) in terms of the highest weight of %. When we say that weight is going to infinity, it means that m(%) grows to to in the usual sense.

The main result on error bounds alluded to above is the following. (See Theorems 9.16 and 9.19 for the precise statements and Remarks 9.18 and 9.21 for an explicit choice of constants.) A uniform bound on orbital integrals, cf. (1.9) below, enters the proof of (ii) [but not (i)].

Theorem 1.3 Let F = F(N, %) be as above. Consider a prime p, an integer k ^ 1, and a function (pp e Hur (G (Qp ))^k such that l4>p | ^ 1 on G (Qp).

(i) (level aspect) Suppose that % remains fixed. There exist constants Alv, B\v, C\v > 0 depending only on G such that for any p, k, (pp as above and for any N coprime to p,

¿FT (¿p) — ¿1,ur (¿p) = O (pAlv+BlvK N—Clv).

(ii) (weight aspect) Fix a level N. There exist constants Awt, Bwt, Cwt > and a lower bound c > 0 depending only on G such that for any p ^ c, k , 4> p as above with (p, N) = 1 and for any %,

¿fT (¿p ) — ¿¿?'Ur (¿p ) = O (pAwt+Bwt k m(%)—Cwt).

Let {Fk = F(Nk, %k)}k>1 be either kind of family in Sect. 1.1, namely either Nk ^ to and %k is fixed or Nk is fixed and %k ^ to. When applied to

{Fk }k^1, Theorem 1.3 leads to the equidistribution results in the following corollary [cf. cases (ii) and (iii) in the paragraph below Theorem 1.1]. Indeed, (i) of the corollary is immediate. Part (ii) is easily derived from the fact that Tp1,ur weakly converges to TST as p Although the unramified Hecke

algebra at p gives rise to only regular functions on the complex variety Tc/ Q, it is not difficult to extend the results to continuous functions on Tc/Q. (See Sects. 9.6-9.8 for details.)

Corollary 1.4 Keep the notation of Theorem 1.3. Let T be a continuous function on Tc/ Q. In view of (1.1) T can be extended by zero to a function Tp on G(Qp)A,ur for each prime p.

(i) (Automorphic Plancherel density theorem [99])

lim TFup(Tp) = T?'ur(Tp).

(ii) (Sato-Tate theorem for families) Let {pk }k^1 be a sequence of primes

tending to m. Suppose that ^k ^ 0 (resp. —°g ^k--> 0) ask ^

log Nk log m(%k)

m if %k (resp. Nk) remains fixed as k varies. Then

lim TFku^Up*) = TT(>T)■

k^m k rk

Theorem 1.3 and Corollary 1.4 remain valid if any finite number of primes are simultaneously considered in place of p or pk. Moreover (i) of the corollary holds true for more general (and possibly ramified) test functions Tp on G(Qp)A thanks to Sauvageot's density theorem. It would be interesting to quantify the error bounds in this generality. Finally the above results should be compared with the proposition 4 in [97] and the theorem 1 in [78] for modular forms on GL(2). We also note [90] for Maass forms (which are not considered in the the present paper).

1.5 Random matrices

We provide a brief account of the theory of random matrices. The reader will find more details in Sect. 11.1 and extensive treatments in [59,74].

The Gaussian unitary ensemble and Gaussian orthogonal ensemble were introduced by Wigner in the study of resonances of heavy nucleus. The Gaussian symplectic ensemble was introduced later by Dyson together with his circular ensembles. In this paper we are concerned with the ensembles attached to compact Lie groups which are introduced by Katz-Sarnak and occur in the statistics of L-functions. (See [39] for the precise classification of these ensembles attached to different Riemannian symmetric spaces.)

One considers eigenvalues of matrices in compact groups G (N) of large dimension endowed with the Haar probability measure. We have three symmetry types G = SO(even) (resp. G = U, G = USp); the notation says that for all N ^ 1, the groups are G(N) = SO(2N) [resp. G(N) = U(N) and G (N) = USp(2 N)].

For all matrices A e G( N) we have an associated sequence of normalized angles

0 ^ #1 < $2 < ••• < $ N < N. (1.3)

For example in the case G = U, the eigenvalues of A e U( N) are given by e(#j) = e2in#j/N for 1 ^ j ^ N. The normalization is such that the mean spacing of the (#i) in (1.3) is about one.

For each N ^ 1 these angles (#i are correlated random variables (a

point process). By the Weyl integration formula their joint density is proportional to

H sin! ' j 1 d $1 ...d #N. (1.4)

1<! < j ^N ^ '

The parameter f is a measure of the repulsion between nearby eigenvalues. We have that f = 1 (resp. f = 2, f = 4) for G = SO(even) (resp. G = U, G = USp).

A fundamental result of Gaudin-Mehta and Dyson, which has been extended to the above ensembles by Katz-Sarnak, is that when N ^to the distribution of the angles (#i converges to a determinantal point process.3 The

kernel of the limiting point process when G = U is given by the Dyson sine kernel

sin n(x — y)

K (x, y) =---—, x, y e R+

n(x — y)

The kernel for G = SO(even) is K+(x, y) = K(x, y) + K(—x, y) and the kernel for G = USp is K—(x, y) = K(x, y) — K(—x, y).

In particular this means that there is a limiting 1-level density W(G) for the angles (#i )1^i as N ^ to (see also Proposition 11.1). It is given by the following formulas:

sin 2n x

W(SO(even))(x) = K+(x, x) = 1 +-,

W(U)(x) = K(x, x) = 1, (1.5)

sin 2n x

W(USp)(x) = K —(x, x) = 1--.

3 For other values of f = 1, 2, 4, the limiting statistics attached to (1.4) has been determined recently by Valko-Virag in terms of the Brownian carousel.

1.6 Low-lying zeros

We can now state more precisely our results on families of L-functions. Let F = r*F be a family of L-functions as defined above in Sects. 1.1-1.2.

For all n e Fk we denote by p j (n), the zeros of the completed L-function A(s, n), where j e Z. We write pj(n) = 2 + iYj(n) and therefore — 1 < Reyj (n) < 1 for all j. By the functional equation A( 1 + iY,n) = 0 if and only if A(2 + iy, n) = 0. We do not assume the GRH that would further imply Yj (Pi) e R for all j.

In the case that n is self-dual the zeros occur in complex pairs, namely L(1 + i y, n) = 0 if and only if A( ± — i y, n) = 0. '

Following Iwaniec-Sarnak we associate an analytic conductor C(Fk) ^ 1 to the family, see Sects. 4.2 and 11.5. We assume from now that the family is in the weight aspect, so that for each k ^ 1, all of n e Fk share the same archimedean factor nm and we can set C(Fk) := C(nm). (For families in the level aspect we obtain similar results, see Sect. 11). Note that C(Fk) ^ m and furthermore we shall make the assumption that log C (Fk) x log m (%k) as k ^m.

For a given n e Fk the number of zeros Yj (n) of bounded height is xlog C(Fk). The low-lying zeros of A(s, n) are those within distance O (1/ log(C(Fk)) to the central point; heuristically there are a bounded number of low-lying zeros for a given n e Fk, although this can only be proved on average over the family. For a technical reason related to the fact that the explicit formula counts both the zeros and poles of A(s, n) (Sect. 4.4), we make an hypothesis on the occurrence of poles of A(s, n) for n e F*, see Hypothesis 11.2.

The statistics of low-lying zeros of the family are studied via the functional

D(Fk; = pF^ Z log C (Fk (1.6)

|Fk 1 UeFk j ^ 2 '

where $ is a Paley-Wiener function. This is the 1-level density for the family Fk. Choosing $ as a smooth approximation of the characteristic function of an interval [a, b], the sum (1.6) should be thought as a weighted count of all the zeros of the family lying in [a, b]:

2an 2bn

-< Yj(n) <-, (j e Z,n e F*).

log C (Fk ^ J log C (Fk) W

We want to compare the asymptotic as k with the limiting 1 -level density of normalized angles (1.3) of the random matrix ensembles described in Sect. 1.5 above.

Theorem 1.5 Let r: LG :^ GL(d, C) be a continuous L-homomorphism which is irreducible and non-trivial when restricted to G. There exists 8 > 0 depending on F such that the following holds. Let F = r*F be a family of L-functions in the weight aspect as in Sect. 1.2, assuming the functoriality conjecture as in Hypothesis 10.1. Assume Hypothesis 11.2 concerning the poles of A(s, n) for n e Fk. Then for all Paley-Wiener functions $ whose Fourier transform $ has support in (—8, 8):

(i) there is a limiting 1-level density for the low-lying zeros, namely there is a density W (x) such that

(ii) the density W (x) is determined by the Frobenius-Schur indicator of the irreducible representation r. Precisely,

The constant 8 > 0 depends on the family F, in other words it depends on the group G, the L-morphism r: LG ^ GL(d, C) and the limit of the ratio

the exponents in the error term occurring in Theorem 1.3. Although we do not attempt to do so in the present paper, it is interesting to produce a value of 8 that is as large as possible, see [53] for the case of GL(2). This would require sharp bounds for orbital integrals as can be seen from the outline below. A specific problem would be to optimize the exponents a, b, e in (1.9). (In fact we can achieve e = 1, see Sect. 1.7 below.)

Our proofs of Theorems 1.3 and 1.5 are effective in the sense that each constant and each exponent in the statements of the estimates could, in principle, be made explicit. Finally we note that, refining the work of E. Royer, Cogdell and Michel [31] have studied the question of distribution of L-values at the edge in the case of symmetric powers of GL(2) and noted in that context the relevance of the indicator s(r).

1.7 Outline of proofs

A wide range of methods are used in the proof. Among them are the ArthurSelberg trace formula, the analytic theory of L-functions, representation theory and harmonic analysis on p-adic and real groups, and random matrix theory.

$(x )W (x )dx;

W(SO(even)), ifs(r) = — 1, W = W (U), ifs (r) = 0,

W (USp), ifs (r) = 1.

log C (Fk) log m(&)

. Its numerical value is directly related to the numerical values of

The first main result of our paper is Theorem 1.3, proved in Sect. 9. We already pointed out after stating the theorem that the Sato-Tate equidistribution for families (Corollary 1.4) is derived from Theorem 1.3 and the fact that the Plancherel measure tends to the Sato-Tate measure as the residue characteristic is pushed to m.

Let us outline the proof of the theorem. In fact we restrict our attention to part (ii), as (i) is handled by a similar method and only simpler to deal with. Thus we consider F with fixed level and weight %, where % is regarded as a variable. Our starting point is to realize that for Tp e C«(G(Qp)), we may interpret Top (Tp) in terms of the spectral side of the trace formula for G evaluated against the function TpT«'pT« e C«(G(A)) for a suitable Tp (depending on F and p; note that p is allowed to vary) and an Euler-Poincare function Tm at m (depending on %). Applying the trace formula, which has a simple form thanks to T«, we get a geometric expansion for T^1)?1 (Tp):

TFT (Tp) = Z Z a^Y • Of (1.8)

M cG y eM (Q)/~ cusp.Levi R—ell

where a'M e C is a coefficient encoding a certain volume associated with the connected centralizer of y in M and T« is the constant term of T« along (a parabolic subgroup associated with) M . The Plancherel formula identifies the term for M = G and y = 1 with Tp?1 (Tp), which basically dominates the right hand side.

The proof of Theorem 1.3 (ii) boils down to bounding the other terms on the right hand side of (1.8). Here is a rough explanation of how to analyze each component there. The first summation is finite and controlled by G, so we may as well look at the formula for each M. There are finitely many conjugacy classes in the second summation for which the summand is nonzero. The number of such conjugacy classes may be bounded by a power of p where the exponent of p depends only on k (measuring the "complexity" of Tp). The term a'M , when unraveled, involves a special value of some Artin L-function. We establish a bound on the special value which suffices to deal with a'M .

The last term dim % can be estimated by using a character formula for the

stable discrete series character (Y, %) as well as the dimension formula

for %. It remains to take care of OM(A \T«). This turns out to be the most difficult task since Theorem 1.3 asks for a bound that is uniform as the residue characteristic varies.

We are led to prove that there exist a, b, e > 0, depending only on G, such that for almost all q,

y ((q)

< qa+bKDM(y)—e/2 (1.9)

for all semisimple y and all (q with (q e Hur(M(Qq))^k and |(q| ^ 1, where DM(■) denotes the Weyl discriminant. The justification of (1.9) takes up the whole of Sect. 7. The problem already appears to be deep for the unit elements of unramified Hecke algebras in which case one can take k = 0. (By a different argument based on arithmetic motivic integration, Cluckers, Gordon, and Halupczok establish a stronger uniform bound with e = 1. This work is presented in Appendix B.) At the (fixed) finite set of primes where wild ramification occurs, the problem comes down to bounding the orbital integral IOM(Qq\(q )| for fixed q and (q. It is deduced from the Shalika germ theory that the orbital integral is bounded by a constant, if normalized by the Weyl discriminant DM(y)1/2, as y runs over the set of semisimple elements. See Appendix A by Kottwitz for details.

We continue with Theorem 1.5. The proof relies heavily on Theorem 1.3. The connection between the two statements might not be immediately apparent.

A standard procedure based on the explicit formula (see Sect. 4) expresses the sum (1.6) over zeros of L-function as a sum over prime numbers of Satake parameters. The details are to be found in Sect. 12, and the result is that D(Fk, $) can be approximated by

y IZkp($pl0gp V (1.10)

^ 'p ((P)\n log C(Fk)

prime p

Here (p e Hur(G(Qp))^k is suitably chosen such that (np) is a sum of powers of the Satake parameters of r* n (see Sects. 2 and 3). The integer k may be large but it depends only on r so should be considered as fixed. Also the sum is over unramified primes. We have log C(Fk) x log m (%k) (see Sects. 10 and 11). We deduce that the sum is supported on those primes p ^ m (%k)A8 where A is a suitable constant and 8 is as in Theorem 1.5.

We apply Theorem 1.3 which has two components: the main term and the error term. We begin with the main term which amounts to substituting $p1,ur ($p) for iFJ ($p) in (1.10). Unlike this term is purely local, thus

simpler. Indeed $p1,ur ($p) can be computed explicitly for low rank groups, see e.g. [48] for all the relevant properties of the Plancherel measure. However we want to establish Theorem 1.5 in general so we proceed differently.

Using certain uniform estimates by Kato [57], we can approximate $pl ur ($p) by a much simpler expression that depends directly on the restriction of r to G x Wq . Then a pleasant computation using the Cebotarev equidis-tribution theorem, Weyl's unitary trick and the properties of the Frobenius-

Schur indicator shows that the sum over primes of this main term contribute —(^ $(0) to (1.10). This exactly reflects the identities (1.7) in the statement (ii) of Theorem 1.5.

We continue with the error term O (pAwt+Bwtkm (%k)—Cwt) which we need to insert in (1.10). We can see the reasons why the proof of Theorem 1.5 requires the full force of Theorem 1.3 and its error term: the polynomial control by pAwt+Bwtk implies that the sum over primes is at most m (%k)DS for some D > 0; the power saving m (%k)—Cwt is exactly what is needed to beat m(%k)DS when S is chosen small enough.

1.8 Notation

We distinguish the letter F for families of automorphic representations on general reductive groups and F = r*F for the families of automorphic representations on GL(d).

Let us describe in words the significance of various constants occurring in the main statements. We often use the convention to write multiplicative constants in lowercase letters and constants in the exponents in uppercase or greek letters.

• Theexponent j from Lemma 2.6 is such that for all T e Hur (GLd) of degree at most k, the pullback r *T is of degree at most ^ jK.

• The exponent be from Lemma 2.14 controls a bound for the constant term ITM(1)1 for all Levi subgroups M and T e Hur(G) of degree at most k.

• The exponent 0 <6 < 1 is a nontrivial bound towards Ramanujan-Petersson for GL(d, A).

• The integer i ^ 1 in Corollary 6.9 is an upper-bound for the ramification of the Galois group Gal(E/F).

• The constants Bs and cs in Lemma 8.4 and A3, B3 in Proposition 8.7 control the number of rational conjugacy classes intersecting a small open compact subgroup.

• The integer uG ^ 1inLemma8.11 is a uniform upper bound for the number of G(F„)-conjugacy classes in a stable conjugacy class.

• The integer nG ^ 0 is the minimum value for the dimension of the unipotent radical of a proper parabolic subgroup of G over F.

• The constant c > 0 is a bound for the number of connected components n0(Z(Iy f ) in Corollary 8.12.

• The exponents Aiv, Biv, Civ > 0 in Theorem 9.16 (see also Theorem 1.3) and Awt, Bwt, Cwt > 0 in Theorem 9.19.

• For families in the weight aspect, the constant n > 0 which may be chosen arbitrary small enters in the condition (11.5) that the dominant weights attached to %k stay away from the walls.

• The exponent Cpole > 0 in the Hypothesis 11.2 concerning the density of poles of L-functions.

• The exponents 0 < C1 < C2 control the analytic conductor C (Fk) of the families in the weight aspect [Inequality (11.7)] and 0 < C3 < C4 in the level aspect (Hypothesis 11.4).

• The constant 8 > 0 in Theorem 11.5 controls the support of the Fourier transform $ of the test function

• The constant c( f) > 0 depending on the test function f is a uniform upper

bound for normalized orbital integrals DG (y) 2 Oy( f) (Appendix A).

Several constants are attached directly to the group G such as the dimension dG = dim G, the rank rG = rk G, the order of the Weyl group wG = | i2|, the degree sG of the smallest extension of F over which G becomes split. Also in Lemma 2.14 the constant bG gives a bound for the constant terms along Levi subgroups. The constants aG, bG, eG in Theorem 7.3 gives a uniform bound for certain orbital integrals. In general we have made effort to keep light and consistent notation throughout the text.

In Sect. 6 we will choose a finite extension E/F which splits maximal tori of subgroups of G. The degree s^pl = [E: F] will be controlled by s^1 ^ sGwG (see Lemma 6.5), while the ramification of E/F will vary. In Sect. 5 we consider the finite extension F1 / F such that Gal( F / F) acts on G through the faithful action of Gal(F1/F). For example if G is a non-split inner form of a split group then F1 = F .In Sect. 12 we consider a finite extension F2/F\ such that the representation r factors through G x Gal( F2/F). For a general G, there might not be any direct relationship between the extensions E/F and F2/F1/ F.

1.9 Structure of the paper

For a quick tour of our main results and the structure of our arguments, one could start reading from Sect. 9 after familiarizing oneself with basic notation, referring to earlier sections for further notation and basic facts as needed.

The first Sects. 2 and 3 are concerned with harmonic analysis on reductive groups over local fields, notably the Satake transform, L-groups and L-morphisms, the properties of the Plancherel measure and the Macdonald formula for the unramified spectrum. We establish bounds for truncated Hecke algebras and for character traces that will play a role in subsequent chapters. In Sect. 4 we recall various analytic properties of automorphic L-functions on GL(d) and notably isobaric sums, bounds towards Ramanujan-Petersson and the so-called explicit formula for the sum of the zeros. Section 5 introduces the Sato-Tate measure for general groups and Sato-Tate equidistribution for Satake parameters and for families. The next Sect. 6 gathers various back-

ground materials on orbital integral, the Gross motive and Tamagawa measure, discrete series characters and Euler-Poincare functions, and Frobenius-Schur indicator. We establish bounds for special values of the Gross motive which will enter in the geometric side of the trace formula.

In Sect. 7 we establish a uniform bound for orbital integrals of the type (1.9). In Sect. 8 we establish various bounds on conjugacy classes and level subgroups. How these estimates enter in the trace formula has been detailed in the outline above.

Then we are ready in Sect. 9 to establish our main result, an automorphic Plancherel theorem for families with error terms and its application to the Sato-Tate theorem for families. The theorem is first proved for test functions on the unitary dual coming from Hecke algebras by orchestrating all the previous results in the trace formula. Then our result is improved to allow more general test functions, either in the input to the Sato-Tate theorem or in the prescribed local condition for the family, by means of Sauvageot's density theorem.

The last three Sects. 10, 11 and 12 concern the application to low-lying zeros. In complete generality we need to rely on Langlands global functoriality and other hypothesis that we state precisely. These unproven assumptions are within reach in the context of endoscopic transfer and we will return to it in subsequent works.

Appendix A by Kottwitz establishes the boundedness of normalized orbital integrals from the theory of Shalika germs. Appendix B by Cluckers-Gordon-Halupczok establishes a strong form of (1.9) with e = 1 by using recent results in arithmetic motivic integration.

2 Satake transforms

2.1 L-groups and L-morphisms

We are going to recall some definitions and facts from [9, §1, §2] and [62, §1]. Let F be a local or global field of characteristic 0 with an algebraic closure F, which we fix. Let WF denote the Weil group of F and set T := Gal(F/F). Let H and G be connected reductive groups over F. Let (B, T, {X a|aeAv) be a splitting datum fixed by T, from which the L-group

lG = G x WF

is constructed. An L-morphism n: LH ^ LG is a continuous map commuting with the canonical surjections LH ^ WF and LG ^ WF such that nl T is a morphism of complex Lie groups. A representation of LG is by definition a continuous homomorphism LG ^ GL(V) for some C-vector space V with dim V < m such that r |g is a morphism of complex Lie groups. Clearly

giving a representation LG ^ GL(V) is equivalent to giving an L-morphism lG ^ LGL(V).

Let f: H ^ G be a normal morphism, which means that f (H) is a normal subgroup of G. Then it gives rise to an L-morphism LG ^ LH as explained in [9, 2.5]. In particular, there is a T-equivariant map Z(G) ^ Z(H), which is canonical (independent of the choice of splittings). Thus an exact sequence of connected reductive groups over F

1 ^ G1 ^ G 2 ^ G 3 ^ 1 gives rise to a T-equivariant exact sequence of C-diagonalizable groups

1 ^ Z($3) ^ Z($2) ^ Z($1) ^ 1.

2.2 Satake transform

From here throughout this section, let F be a finite extension of Qp with integer ring O and a uniformizer m .Set q := O/m Ol Let G be an unramified group over F and B = TU be a Borel subgroup decomposed into the maximal torus and the unipotent radical in B. Let A denote the maximal F-split torus in T. Write $F (resp. $) for the set of all F-rational roots (resp. all roots over F) and (resp. ) for the subset of positive roots. Choose a smooth reductive model of G over O corresponding to a hyperspecial point on the apartment for A. Set K := G(O). Denote by X*(A)+ the subset of X*(A) meeting the closed Weyl chamber determined by B, namely X e X*(A)+ if a(X) ^ 0 for all a e . Denote by ^F (resp. the F-rational Weyl group for (G, A) (resp. the absolute Weyl group for (G, T)), and pF (resp. p) the half sum of all positive roots in (resp. $+). A partial order ^ is defined on X*(A) (resp. X*(T)) such that ^ ^ X if X — ¡i is a linear combination of F-rational positive coroots (resp. positive coroots) with nonnegative coefficients. The same order extends to a partial order ^r on X*(A) ®Z R and X*(T) ®Z R defined analogously.

Let Fur denote the maximal unramified extension of F. Let Fr denote the geometric Frobenius element of Gal(Fur/F). Define W^r to be the unramified Weil group, namely the subgroup FrZ of Gal(Fur/F). Since Gal(F/F) acts on G through a finite quotient of Gal( Fur / F), one can make sense of LGur := G x WFr.

Throughout this section we write G, T, A for G(F), T(F), A(F) if there is no confusion. Define Hur(G) := Cf (K\G/K) and Hur(T) := C™(T(F)/T(F) n K). The latter is canonically isomorphic to Hur(A) := Cf( A( F)/A(O)) via the inclusion A T .We can further identify

Hur(T) — Hur(A) - C[X*(A)]

where the last C-algebra isomorphism matches X e X *(A) with 1^(^)(AnK) e Hur(A). Let X e X*(A). Write

tX := 1KX(rn)K e H"r(G), rA := —— Z '1wX(m)(AnK) e (A)^F

1 F 1 we^F

The sets {t^Xx,(A)+ and {taA}xexa)+ are bases for Hur(G) and Hur(A)^F as C-vector spaces, respectively. Consider the map

Hur(G) ^ Hur(T), f ^ ^t ^ 5b(t)1/2 J f (tu)du^ (2.1)

composed with Hur(T) — Hur(A) above. The composite map induces a C-algebra isomorphism

SG : Hur(G) Hur(A)^F (2.2)

called the Satake isomorphism. We often write just S for SG. We note that in

general S does not map t g to tA .

Another useful description of Hur(G) is through representations of LGur. (The latter notion is defined as in Sect. 2.1). Write (G x Fr)ss-conj for the set of G-conjugacy classes of semisimple elements in G x Fr. Consider the set

ch Gur) := tr r: (G x Fr)ss-conj ^ C|r is a representation of LGur

Define C[ch(LGur)] to be the C-algebra generated by ch(LGur) in the space of functions on (G x Fr)ss-conj. For each X e X*(A)+ define the quotient

Iwe^F sgn(w)w(X + PF) Xx := —^-—-, (2.3)

l^weiiF sgn(w)wPF

which exists as an element of C[X*(A)]^F and is unique. (One may view xX as the analogue in the disconnected case of the irreducible character of highest weight X, cf. proof of Lemma 2.1 below.) Then jxX}XeX*(A)+ is a basis for C[X*(A)]^F as a C-vector space, cf. [57, p. 465]. (Another basis was given by tA's above.) There is a canonical C-algebra isomorphism

T: C[ch(LGur)] Hur(A)iiF, (2.4)

determined as follows (see [9, Prop 6.7] for detail): for each irreducible r, tr r |$ is shown to factor through $ ^ A (induced by A c T). Hence tr r |$ can be viewed as an element of C[X*(Ai)] = C[X*(A)], which can be seen to be invariant under QF. Define T(tr r) to be the latter element.

Let r0 be an irreducible representation of G$ of highest weight X0 e X*($)+ = X*(T)+. The group W^ acts on X*($)+. Write Stab(X0) c W^r for the stabilizer subgroup for X0, which has finite index (since a finite power of Fr acts trivially on G and thus also on $). Put r := Ind-3" w, r0 and

J y G xStab(X0) 0

X := XaeWur/Stab(Xo)ffXo e X*(A)+. Clearly r and X depend only on the WFr-orbitof X0. Puti(X0) := [WF: Stab(X0)].

Lemma 2.1 (i) Suppose that r and X are obtained from r0 and X0 as above. Then

T (tr r) = xx- (2.5)

(ii) In general for any irreducible representation r : LGur ^ GLd (C) such

that r'(WF*) has relatively compact image, let r0 be any irreducible sub-

representation ofr '1$. Letr be obtained from r0 as above. Then for some Z e Cx with IZ 1 = 1,

tr r' = Z • tr r.

Proof Let us prove (i). For any i ^ 1, let LGi denote the finite L-group G x Gal( Fi / F) where Fi is the degree i unramified extension of F in F.

It is easy to see that r(Fr!(X0)) is trivial and that r = IndLG'(X0}r0. Then (2.5) amounts to Kostant's character formula for a disconnected group [61, Thm 7.5] applied to lG'(X0). As for (ii), let X0 and X be as in the paragraph preceding the lemma. Let j ^ 1 be such that G becomes split over a degree j unramified extension of F. (Recall that G is assumed to be unramified.) By twisting r' by a unitary character of W^r one may assume that r' factors through LG j. Then

Lq ■

both r and r' factor through L G j and are irreducible constituents of Ind- 1 r0. From this it is easy to deduce that r' is a twist of r by a finite character of W™ of order dividing j. Assertion (ii) follows. □

Each X e X*(A)+ determines sXe C such that

5—11(xx) = X Sx^tq (2.6)

¡e X*( A)+

where only finitely many sX,/A, are nonzero. In fact Theorem 1.3 of [57] identifies sX,/A, with KXl(q—1) defined in (1.2) of that paper, cf. §4 of [48]. In particular

sx, x = 0 and sx x = 0 unless / ^ X . The following information will be useful in Sect. 2.7.

Lemma 2.2 LetX, / e X*(A)+.SupposethatX*w/x := w(X+pF) — (/x+pF) is nontrivial for all w e QF. For k e X*(A) let p(k) e be the number of tuples (cav )ave(&v )+ with cav e such thatX av cav • av = k . Then

|sx^ q-1|^F I max p(X*w /).

Proof It is easy to see from the description of Kx,x(q-1) in [57, (1.2)] that

K¡(q-1 )| ^ I^fI max P (w(X + pf) - (/ + Pf); q-1) • we^F

The definition of PMn [57, (1.1)] shows that 0 ^ P(k; q-1) ^ p(K)q-1 if k = 0. □

2.3 Truncated unramified Hecke algebras

Set n := dim T and X*(T)R := X*(T) ®Z R. Choose an R-basis B = (eien} of X*(T)R. For each X e X*(T)R, written as X = Xn=i a' (X)e' for unique ai (X) e R, define

|X|B := max |ai(X )|, ||X||B := max(|<wX\B).

i^i ^n

When there is no danger of confusion, we will simply write |-|B or even | ■ | instead of | ■ |B, and similarly for || ■ ||B. It is clear that || ■ ||B is ^-invariant and that |X1 + X 2|B ^ |X 1 |B + |X 2|B for all X 1, X 2 e X*(T).When k e define

Hur (G)

C-subspace of Hur(G) generated by tG, X e X *(A)+,

\b ^ K

It is simply written as Hur(Gwhen the choice of B is clear.

Lemma 2.3 Let B and B' be two R-bases of X*(T)r. Then there exist constants c1, c2, Bi, B2, B3, B4 > 0 such that for all X e X*(T )r,

(i) ci|X|ß/ ^ |X|ß ^ c2|X|B,

(ii) Bi|X|B < ||X|B < B2XB for all X e X *(T)r,

(iii) B3||X||B/ < ||x|b < B4|X|B/ for all X e X*(T)r and

(iv) Hur(G)^b-1k>b' c Hur(G)^k'b c Hur(G)^b-1k'b'.

Proof Let us verify (i). As the roles of B and B' can be changed, it suffices to prove the existence of c2. For this, it suffices to take c2 = sup^ IXIB. The latter is finite since | • IB is a continuous function on the set of X such that IXIB ^ 1, which is compact. Part (ii) is obtained by applying the lemma to the bases B' = mB for all m e i2. Let us check (iii). Let B1, B2 > 0 (resp. B1, B2 > 0) be the constants of (ii) for the basis B (resp. B'). Then

c1 B1(B2)—1\\X\y < c1 B1IXB < B1IXIB < \\X\\s

and similarly \\X\\B ^ c2B2(B1)—1\X\B/. Finally (iv) immediately follows from (iii). □

It is natural to wonder whether the definition of truncation in (2.7) changes if a different basis {tQ} or {xx} is used. We assert that it changes very little in a way that the effect on k is bounded by a k -independent constant. To ease the statement define HU(G)^k,b for i = 1 (resp. i = 2) to be the C-subspace of Hur(G) generated by S—1(t£) (resp. S—1(xx)) for X e X*(A)+ with \\X\\B ^ k.

Lemma 2.4 There exists a constant C ^ 1 such that for every k e and for any i, j e{0,1, 2},

Hur(G)^k'b c Hur(G)^Ck'b-

Proof It is enough to prove the lemma for a particular choice of B by Lemma 2.3. So we may assume that B extends the set of simple coroots in by an arbitrary basis of X*(Z(G))R. Again by Lemma 2.3 the proof will be done if we show that each of the following generates the same C-subspace:

(i) the set of tQQ for X e X*(A)+ with IXIB ^ k,

(ii) the set of S—1(t£) for X e X*(A)+ with IXB < k,

(iii) the set of S—1(xx) for X e X*(A)+ with IXB < k.

It suffices to show that the matrices representing the change of bases are "upper triangular" in the sense that the (X, X) entries are nonzero and (X, ¡i) entries are zero unless X ^ (Note that X ^ ¡i implies IXIB ^ ¡IB by the choice of B.) We have remarked below (2.3) that sX,^'s have this property, accounting for (i)^(iii). For (ii)^(iii) the desired property can be seen directly from (2.3) by writing xX in terms of tA's. □

2.4 The case of GLd

The case G = GLd is considered in this subsection. Let A = T be the diagonal maximal torus and B the group of upper triangular matrices. For 1 ^ i ^ d,

take Y' e X*(A) to be y ^ diag(1,.1, y, 1,---, 1) with y in the i-th place. One can naturally identify X*(A) ~ Zd such that the images of Y form the standard basis of Zd. Then QF is isomorphic to S, the symmetric group in d variables acting on {Y1,-.., Y2} via permutation of indices. We have the Satake isomorphism

S: Hur (GLd) ^ Hur (T )Qf - C [Y±, Y±]Sd .

For an alternative description let us introduce standard symmetric polynomials X1,-.., X2 by the equation in a formal Z-variable (Z — Y1)...(Z — Y2) = Zd — X1 Zd—1 + ••• + (—1)dXd. Then

C [Y±,---,Y±fd = C [ X1,--., Xd—1, X±] -

Let k e Z^0. Define Hur(GL2)^k, or simply HfK, to be the preimage under S of the C-vector space generated by

Z Y%)Y7m---Y^): a1,---,ad e[—K,K] -

The following is standard (cf. [48]).

Lemma 2.5 Let r e Z>1. Let Xr := (r, 0, 0,---, 0) e X*(A)+. Then S~\Y[ + • •• + Yrd) = X cXri • tQ

le X*(A)+

for cXr e C with cXr ,Xr = qr (1 d)/2, where the sum runs over the set of l e X +(T)+ such that ¡i ^ Xr. In particular,

S—1(Y1 + ••• +Yd) = q(1—d ^tQ

S—1 Y2 + ••• + y2) = q1—d (tq^.,0) + (1 — q)T(Q,1,0,...,0)) -

(1,0,---,0)' 0,

2.5 L-morphisms and unramified Hecke algebras

Assume that H and G are unramified groups over F. Let n: LH ^ LQ be an unramified L-morphism, which means that it is inflated from some L-morphism L Hur ^ LGur (the notion of L-morphism for the latter is defined as in Sect. 2.1). There is a canonically induced map ch(LGur) ^ ch(LHur). Via

(2.2) and (2.4), the latter map gives rise to a C-algebra map n*: Hur(G) ^ Hur (H).

We apply the above discussion to an unramified representation

r: lG ^ GLd(C).

Viewing r as an L-morphism LG ^ LGLd, we obtain

r*: Hur(GLd) ^ Hur(G).

Lemma 2.6 Let B be an R-basis of X*(T)r. There exists a constant ff > 0 (depending on B, d and r) such that for all k e , r*(Hur(GLd)^k) c Hur(G.

Proof Thanks to Lemma 2.3, it is enough to deal with a particular choice of B. Choose B by extending the set Av of simple coroots, and write B = Av ]J B0. We begin by proving the following claim: let X1, X2 e X*(A)+ and expand the convolution product

T G* - G_^ap G

TX1 * TX2 = Z^I aX 1, X2 Tp p

where only p e X*(A)+ such that p X 1 + X2 contribute (cf. [18, p. 148]). Only finitely many terms are nonzero. Then the claim is that

|p|b ^ |X 1 + X 2|b, whenever ap1 M = 0.

To check the claim, consider p = XeeB ae (p) ■ e and X 1 + X2 = XeeB ae (X1 + X2) ■ e, where the coefficients are in R. The conditions p X 1 + X2 and p e X*(T)R,+ imply that ae(p) = ae(X 1 + X2) if e e B0 and 0 ^ ae(p) ^ ae(X 1 + X2) if e e Av. Hence |p|B ^ |X 1 + X2|B.

We are ready to prove the lemma. It is explained in Lemma 2.4 and the remark below it that there exists a constant f 1 > 0 which is independent of k such that every $ e Hur(GLd)^k can be written as a C-linear combination of

Z YO(1)C) • • • K(dy a1ad e [-f 1k, f 1k].

Each element above can be rewritten in terms of the symmetric polynomials Xi's of Sect. 2.4: first, Xf1K times XO eSd YO\1) YO2(2) • • • Yaa\d) is a symmetric polynomial of degree ^ 2f 1k , which in turn is a polynomial in XXd of degree ^ 2f 1k. We conclude that every $ e Hur(GLd)^k is in the span of monomials

Xb1 X b2... Xbdd, bu...,bd e[-2faK, 2&k ]. (2.8)

For each 1 ^ i ^ d, write r*(Xi) [resp. r* (X-1)] as a linear combination

of tg (resp. tG ) with nonzero coefficients. Define f0 to be the maximum

Xhi hi

among all possible |Xi, j | and |X- j |. The above claim r *( X b1 X b2... Xb/) as in (2.8) is in the C-span of tG satisfying

Hb < (|b1| + ••• + bd|)fo < 2dp0p1K.

So the above span contains r*(0) for 0 e Hur(GLd)^k. By Lemma 2.3 there exists a constant B2 > 0 such that \\iA\b ^ B2|^|B for every ^ e X*(T). Hence the lemma holds true with f := 2B2df0f1. □

The map r also induces a functorial transfer for unramified representations

r*: Irrur(G(F)) ^ Irrur(GLd(F)) (2.9)

uniquely characterized by tr r*(n)(0) = tr n(r*0) for all n e Irrur(G(F)) and 0 e Hur(GLd(F)).

2.6 Partial Satake transform

Keep the assumption that G is unramified over F. Let P be an F-rational parabolic subgroup of G with Levi M and unipotent radical N such that B = TU is contained in P. Let QM (resp. i2MtF) denote the absolute (resp. F-rational) Weyl group for (M, T). A partial Satake transform is defined as [cf. (2.1)]

SM: Hur(G) ^ Hur(M), f ^ (m ^ Sp(m)1/2 J f (mn)dn^

It is well known that SG = SM o SM. More concretely, SM is the canonical inclusion C[X*(A)]^M,F ^ C[X*(A)]^F if Hur(M) and Hur(G) are identified with the source and the target via SG and SM , respectively. Since T is a common maximal torus of M and G, an R-basis B of X*(T)R determines truncations on Hur(M) and Hur(G).

Lemma 2.7 For any k e Z^o, SM(Hur(G)^k'b) c Hur(M)^k'b.

Proof It is enough to note that \\X\\B,M ^ \\X\\B;G for all X e X*(A), which holds since the QM-orbit of X is contained in the ^-orbit of X . □

L H/T v L,

Remark 2.8 Let n: LM ^ LG be the embedding of [9, §3], well defined up

to G-conjugacy. Then SM coincides with n* : Hur (G) ^ Hur (M) of Sect. 2.5

2.7 Some explicit test functions

Assume that r: LG = G x WF ^ GLd (C) is an irreducible representation arising from an unramified L-morphism LGur ^ LGL^r such that r(Wf) is relatively compact. For later applications it is useful to study the particular element r*Y + ••• + Yd) in Hur(G).

Lemma 2.9 Let $ = r*(Y1 +-----+Yd). Then

(i) Suppose that r: LGur ^ GLd (C) does not factor through WF* (orequiv-alently that r |g is not the trivial representation). Then

$(1^ < f| max p(X*w 0) ■ q-1.

(ii) Suppose thatr G is trivial. Then $(1) = r (Fr).

Proof Let us do some preparation. By twisting r by an unramified unitary character of WF (viewed as a character of LG) we may assume that r =

lg ■ ^

Indg ]r0 for some irreducible representation r0 of G, cf. the proof of Lemma

2.1 (ii). Let X0 be the highest weight of r0 and define X e X*(A)+ as in the

paragraph preceding Lemma 2.1. The lemma tells us that S($) = ZxX e

C[X*(A)]"F with Z| = 1.

In the case of (ii), r is just an unramified unitary character of WF (with

d = 1), and it is easily seen that xX = tA, Z = r (Fr), and so $(1) = r(Fr).

Let us put ourselves in the case (i) so that X = 0. Note that $(1) is just the

coefficient of rf when $ = ZS-1(xX) is written with respect to the basis

{tG}. Such a coefficient equals Z ■ sX,0 according to (2.6), so |$(1)| = |sX,0|.

Now Lemma 2.2 concludes the proof. [Observe that X*w 0 = 0 whenever

0 = X e X*(A)+.] □

2.8 Examples in the split case

When G is split, it is easy to see that C[ch(LGur)] is canonically identified with C[ch(G)] which is generated by finite dimensional characters in the space of functions on G. So we may use C[ch(G)] in place of C[ch(LGur)].

Example 2.10 (When G = Sp2n, n ^ 1)

Take r: G = SO2n+1 (C) ^ GL2n+1 (C) to be the standard representation. Then

Y1 + ••• + Y2„+1 = tr (Std) e C[ch(GL2„+1)]

is mapped to tr (r) e C[ch(SO2n+1)] and

Y12 + ••• + Y^n+1 = tr (Sym2(Std) - A2(Std)) e C[ch(GL2„+i)]

is mapped to tr (r) e C[ch(SO2n+1)]. Then Sym2(V) breaks into C and an irreducible representation of G of highest weight (2, 0,..., 0) in the standard parametrization. When n > 1, a2(V) is irreducible of highest weight (1, 1, 0,..., 0). When n = 1, a2(V) - Vv, i.e. isomorphic to (Std)v. (See [41, §19.5].) Let us systematically write AX for the irreducible representation of SO2n+1 with highest weight X. Then

r *(Y; + --- + Y2„+1) = tr A(1,0,...,0),

r* (Y2 + ... + Yl+1) = tr (C + A(2,0,...,0) - A(1,1,0,...,0)). (2.10)

if n ^ 2. If n = 1, the same is true if A(1;1,o,...,o) is replaced with A(-1). For i = 1 , 2, define

$(i> := S-1 (r* (Y1 + ..• + Y^)) . Then one computes

(1) 1—2n

$ ; = q 2 IkiH1A...fi)(mv)K,

$(2) = Ik + q1—2n Ik ^...M^k — q1—2n (q — 1)1 k p(1X0,...M^v)K ■

where /xX is the cocharacter of a maximal torus given by X in the standard parametrization. In particular, $(1)(1) = 0 and $(2)(1) = 1.

Example 2.11 (When G = SO2n, n ^ 2)

Take r: G = SO2n (C) ^ GL2n (C) to be the standard representation. Similarly as before, Sym2(V) breaks into C and an irreducible representation of G of highest weight (2, 0,..., 0). When n > 1, a2(V) is irreducible of highest weight (1, 1, 0,..., 0). When n = 1, A2(V) - C. (See [41, §19.5].) The same formulas as (2.10) hold in this case. Defining

$(i] := S—1 (r* (Y; + ••• + Y2„)) , (2.11)

we can compute $(1), $(2) and see that $(1)(1) = 0 and $(2)(1) = 1.

Example 2.12 (When G = SO2n+;)

Take r: G = Sp2n (C) ^ GL2n (C) to be the standard representation. Then

Yi + ••• + Y2n = tr (Std) e C[ch(GL2„)]

is mapped to tr (r o Std) e C[ch(Sp2n)] and Then

Y2 + ... + Y22n = tr (Sym2(Std) - A2(Std)) e C[ch(GL2n)]

is mapped to tr (r o Std) e C[ch(Sp2n)]. If n ^ 2 then a2(V) breaks into C and an irreducible representation of G of highest weight (1, 1, 0,..., 0). (See [41, §17.3].) We have

r *(Y1 + --- + Y2n+1) = tr A(1,0,...,0), r* (Y2 + ••• + Y2n+1) = tr (A(2,0,...,0) - Aa,1,0,...,0) - C).

As in Example 2.10, A designates a highest weight representation (now of Sp2n).Define 0(i ^ as in (2.11). By a similar computation as above, 0(1)(1) = 0, 0(2)(1) = -1.

2.9 Bounds for truncated unramified Hecke algebras

Let F, G, A, T and K be as in Sect. 2.2. Throughout this subsection, an R-basis B of X*(T)r will be fixed once and for all. Denote by p e X*(T) ®z 1Z half the sum of all a e

Lemma 2.13 For any \ e X*(A), [K\i(m)K : K] < qdG +rG

Proof Let vol denote the volume for the Haar measure on G (F) such that vol(K) = 1. Let I c K be an Iwahori subgroup of G(F). Then I = (I n U)(I n T)(I n U). We follow the argument of [106, pp. 241-242], freely using Waldspurger's notation. Our I, U, U, and T will play the roles of his H, U0, U0 and M0, respectively. For all m e M + (in his notation), it is not hard to verify that c'Uq (m) = cu0 (m) = cM0 (m) = 1. Then Waldspurger's argument shows

vol(K\(m)K) < [K : I]2vol(I\(m)I) < [K : I]2qvol(I) = [K : I]q{p'\}.

Finally observe that [K: I] < G(Fq)| < qdG(1 + 1 )rG < qdG +rG. (The

middle inequality is easily derived from Steinberg's formula. cf. [47, (3.1)].)

The following lemma will play a role in studying the level aspect in Sect. 9.

Lemma 2.14 Let M be an F-rational Levi subgroup of G. There exists a constant be > 0 (depending only on G) such that for all k e Z>0 and all 0 e Hur(G)^k'b such that 0 < 1, we have 0m(1)| = O(qdG +rG +bGk) (the implicit constant being independent of k and 0), where we put 0M := SM (0).

Proof When M = G, the lemma is obvious (with bG = 0). Henceforth we assume that M c G .In view of Lemma 2.3, it suffices to treat one R-basis B. Fix a Z-basis {e;,..., edim A} of X*(A), and choose any B which extends that Z-basis. It is possible to write

$ = Z a\ ■ 1k\(m)K

for |a\ ^ 1. Thus

|$M (1)| =

in ( f )

$(n)dn

|\||<k

in ( f )

1k fi(m)K (n)dn

For each K\(m)K is partitioned into left K-cosets. On each coset yK,

/ 1y k (n)dn In (f )

Hence, together with Lemma 2.13,

< vol(K n N(F)) = 1.

|$M(1)| < Z [K\(^)K : K] ^ ^ qdG +rG+(p\

ii€k

Write b0 for the maximum of |(p, ei )| for i = 1,..., dim A. Take bG := b0 dim A + 2dim A. If ||\|| ^ k then \ = XdmA aiei for ai e Z with — k ^ ai ^ k . Hence the right hand side is bounded by (2k + 1)dim AqdG +rG +b0K dim A ^ qdG +rG +bGK since 2k + 1 ^ 22k ^ q

An elementary matrix computation shows the bound below, which will be used several times.

Lemma 2.15 Let s = diag(s;,... , sm ) e GLm ( F v ) and u = (uij)m■=1 e GLm (F v). Define vmin (u) := mini; j v(uij) and similarly vmin (u—;). Then for any eigenvalue X ofsu e GLm (Fv),

v(X) e

vmin (u) + min v(si), — vmin (u 1) + max v (si) ii

Remark 2.16 The lemma will be typically applied when u e GLm (Ov) where Ov is the integer ring of Fv. In this case vmin(u) = vmin(u~l) = 0.

Proof Let V be the underlying Fv-vector space with standard basis {e1,..., em}. Let Bj = {i = (i1, ...,ij)|1 ^ i1 < ■ ■■ < ij ^ m}. Then AjV has a basis {ei1 A ■■■ A eij }ieBj. We claim that

v(tr (su Aj V)) ^ j ■ min v(si).

Let us verify this. Let (u^ )i)i'eBj denote the matrix entries for the u-action on a j V with respect to the above basis. Observe that v(ui,i>) ^ j ■ vmin (u) for all i, i'e Bj .Then

v(tr (su| Aj V)) = v sij

sii sin . . . sij ■ ui,i

^ minv(si1 si2 ...sij ■ ui,i) ^ j ■ minv(si) + minv(uiti) i i i

^ j (min v(si) + vmin (u)). i

The coefficients of the characteristic polynomial for su e GLm (Fv) are given by tr (su | Aj V) up to sign. The above claim and an elementary argument with the Newton polygon show that any root X satisfies v(X) ^ vmin (u)+mini v(si). Finally, applying the argument so far to s -1 and u -1, we obtain the upper bound

for v(X). □

As before, the smooth reductive model for G over O such that G(O) = K will still be denoted G.

Lemma 2.17 Let S: G GLm be an embedding of algebraic groups over O. Then there exists aGLm (O)-conjugate of S whichmaps A (afixedmaximal split torus of G) into the diagonal maximal torus of GLm.

Proof Note that the maximal F-split torus A naturally extend to A c G over O, cf. [103, §3.5]. The representation of A on a free O-module of rank m via S defines a weight decomposition of Om into free O-modules. Choose any refinement of the decomposition to write Om = L1 © ■■■ © Lm, as the direct sum of rank 1 free O-submodules. Let vi be an O-generator of Li for 1 ^ i ^ m. Conjugating S by the matrix representing the change of basis from {v1,...,vm} to the standard basis for Om, one can achieve that S( A) lies in the diagonal maximal torus. □

Let y e G (F) be a semisimple element and choose a maximal torus TY of G defined over F such that y e Ty (F). Denote by $ (G, Ty) the set of roots for Ty in G.

Lemma 2.18 Suppose that there exists an embedding of algebraic groups S : G ^ GLm over O. There exists a constant B5 > 0 such that for every k e Z^0, every \ e X*(A) satisfying ||\|| ^ k, every semisimple y e K\i(m)K and every a e $Y (for any choice of Ty as above), we have — B5K ^ v(a(y)) ^ B5K. In particular, |1 — a(y)| ^ qB5K.

Remark 2.19 Later S will be provided by Proposition 8.1.

Proof We may assume that S(A) is contained in the diagonal torus of GLm, denoted by T, thanks to Lemma 2.17. Write T for the maximal torus of G which isthecentralizerof A so that S(T) c T.Wehaveasurjection X*(T) ^ X* (T) induced by S. For each a in the set of roots $ (G, T), we fix a lift a e X* (T) once and for all. Set c; := maxae$(G,T) ||a||GLm.

Let c2 := max^H^ || S o \||GLm where \ e X*(A)R runs over elements with ||\|| ^ 1. Then for any k e Z^0, ||\|| ^ k implies || S o \||GLm ^ c2k. Hence S (\(m)) is a diagonal matrix in which each entry x satisfies —c2k ^ v(x) ^ c2K.

We can write y = k1\(m)k2 for some k;, k2 e G(O). Then S(y) = ki S(\(m))k'2 for ki, k2 e GLm(O), and S(y) is conjugate to S(\(m)) k2(ki)—It follows from Lemma 2.15 that for every eigenvalue X of S(y), we have —c2k ^ v(X) ^ c2k.

Choose any Ty as above. There exists an isomorphism T — Ty over F induced by a conjugation action t ^ gtg—1 given by some g e G(F). The isomorphism is well defined only up to the Weyl group action but induces a bijection from $(G, T) onto $(G, Ty). Put Ty := S(g)TS(g)—The conjugation by S (g) induces an isomorphism T — Ty over F and a bijection from $(GLm, T) onto $(GLm, TY).LetaY e $(G, Ty) (resp. ay e $(GLm, Ty)) denote the image of a (resp. a) under the bijections. By construction, the composition TY — T ^ T — TY coincides with the restriction of S to TY. Hence the induced map X*(Ty) ^ X*(Ty) maps cxy to aY.

Using the isomorphisms Ty(F) — T(F) — (Fx)m, let (X;,...,Xm) e (Fx)m be the image of S(y) under the composition isomorphism. We may write aY as a character (Fx)m ^ Fx given by (t1,...,tm) ^ t^1 ... ffi with a1,...,am e Z such that —c; ^ ai ^ c; for every 1 ^ i ^ m. We have

aY(y) = aY(s(y)) = Xa1 ■■■Xamm,

so v(aY(y)) = Xm=i aiv(Xi). Hence —mc1c2K ^ v(aY(y)) ^ mcic2k,

proving the first assertion of the lemma. From this the last assertion is obvious.

Remark 2.20 Suppose that F runs over the completions of a number field F at non-archimedean places v, that G over F comes from a fixed reductive group

G over F, and that S comes from an embedding G GLm over the integer ring of F (at least for every v where G is unramified). Then B5 of the lemma can be chosen to be independent of v (and dependent only on the data over F). This is easy to see from the proof.

3 Plancherel measure on the unramified spectrum

3.1 Basic setup and notation

Let F be a finite extension of Qp. Suppose that G is unramified over F. Fix a hyperspecial subgroup K of G. Recall the notation from the start of Sect. 2.2. In particular i2 (resp. i2F) denotes the Weyl group for (Gy, Ty) [resp. (G, A)]. There is a natural Gal (F / F)-actionon under which ^Gal(F/F) = ^ F .(See [9, §6.1].) Since G is unramified, Gal(F/F) factors through a finite unramified Galois group. Thus there is a well-defined action of Fr on i2, and ^Fr = QF.

The unitary dual G(F)A of G(F), or simply GA if there is no danger of ambiguity, is equipped with Fell topology. (This notation should not be confused with the dual group G). Let GA,ur denote the unramified spectrum in GA, and GA,ur,temp its tempered sub-spectrum. The Plancherel measure 00pl on GA is supported on the tempered spectrum GA,temp. The restriction of 00pl to GA'ur will be written as 0pl,ur. The latter is supported on GA'ur'temp. Harish-Chandra's Plancherel formula (cf. [106]) tells us that 00pl(0) = 0(1) for all 0 e H(G(F)). In particular, 0pl'ur(0) = 0(1) for all 0 e Hur(G(F)).

3.2 The unramified tempered spectrum

An unramified L-parameter Wy ^ LGur is defined to be an L-morphism LHur ^ LGur (Sect. 2.5) with H = {1}. Two such parameters f1 and f2 are considered equivalent if f1 = gf2g-1 for some g e G. Consider the following sets:

(i) Irreducible unramified representations n of G (F) up to isomorphism.

(ii) Group homomorphisms x : T(F)/T(F) n K ^ Cx up to i2F-action.

(iii) Unramified L-parameters f : W^r ^ LGur up to equivalence.

(iv) Elements of (G x Fr)ss-conj; this set was defined in Sect. 2.2.

(v) i2Fr-orbits in 0/(Fr - id)0.

(vi) Qf-orbits in A.

(viii) C-algebra morphisms 9 : Hur(G) ^ C.

Let us describe canonical maps among them in some directions. • (i) ^ (vii) Choose any 0 = v e nK. Define 9(0) by 9(0)v = fa (f )0(g)n(g)vdg.

• (ii) ^ (i) n is the unique unramified subquotient of n-ind^F) x.

• (ii) ^ (vi) Induced by Hom(T ( F )/T ( F )n K, Cx) - Hom( A( F )/A( F )n

K, Cx)

- Hom(X*(A), Cx)-Hom(X*(A), Cx)-X* (A) ®z Cx- A (3.1)

where the second isomorphism is induced by X*(A) ^ A(F) sending / to /(m).

• (iii) ^ (iv) Take ^(Fr).

• (v) ^ (iv) Induced by the inclusion t ^ t x Fr from T to G x Fr.

• (v) ^ (vi) Induced by the surjection T ^ T which is the dual of A ^ T. (Recall QFr = Qf .)

• (vii) ^ (vi) Via S : Hur(G) - C[X*(A)]Qf, 9 determines an element of [cf. (3.1)]

Qf\Hom(X*(A), Cx) - QF\A.

Lemma 3.1 Under the above maps, the sets corresponding to (i)—(vii) are in bijection with each other.

Proof See §6, §7 and §10.4 of [9]. □

Let F' be the finite unramified extension of F such that Gal(F/F) acts on G through the faithful action of Gal(F'/F). Write LGF/F := G x Gal(F'/F). Let K be a maximal compact subgroup of G which is Fr-invariant. Denote by Tc (resp. Ac) the maximal compact subtorus of T (resp. A).

Lemma 3.2 The above bijections restrict to the bijections among the sets consisting of the following objects.

(i)t irreducible unramified tempered representations n of G(F) up to isomorphism.

(ii)t unitary group homomorphisms x : T(F)/T(F) n K ^ U (1) up to Qf -action.

(iii)t unramified L-parameters f: WF* ^ LGur with bounded image up to equivalence.

(iv)t G-conjugacy classes in K x Fr (viewed in LGF/F).

(iv)t K-conjugacy classes in K x Fr (viewed in K x Gal(F'/F)).

(v)t ^Fr-orbits in Tc/(Fr — id)Kc.

(vi)t QF-orbits in Ac.

[The boundedness in (iii)t means that the projection of Im f into LGFyF is contained in a maximal compact subgroup ofLGF'/F.]

Proof (i)t ^ (ii)t is standard and (iii)t ^ (iv)t is obvious. Also straightforward is (ii)t ^>(vi)t in view of (3.1).

Let us show that (v)t ^ (vi)t. Choose a topological isomorphism of complex tori 0 ~ (Cx)d with d = dim T. Using Cx ~ U(1) x R^, we can decompose T = Tc x Tnc such that Tnc is carried over to (RX0)d under the isomorphism. The decomposition of T is canonical in that it is preserved under any automorphism of T. By the same reasoning, there is a canonical decomposition A = Ac x Anc with Anc ~ (Rx0)dim A. The canonical surjection T ^ T carries Tc onto tc and Tnc onto Tnc. [This reduces to the assertion in the case of Cx, namely that any maps U(1) ^ Rx0 and Rx0 ^ U(1) induced by an algebraic map Cx ^ Cx of C-tori are trivial. This is easy to check.] Therefore the isomorphism T/(Fr — id) T ^ A of Lemma 3.2 induces an isomorphism Tc/(Fr — id)Tc ^ Ac (as well as Tnc/(Fr — id)Tnc ^ Anc).

Next we show that (iv)t ^ (v)t. It is clear that t ^ t x Fr maps (v)t into (iv)t. Since (v)t and (iv)t are the subsets of (v) and (iv), which are in bijective correspondence, we deduce that (v)t ^ (iv)t is injective. To show surjectivity, pick any k e K. There exists t e T such that the image of t in (iv) corresponds under (iv) ^ (v) to the (^conjugacy class of k x Fr. It is enough to show that we can choose t e Tc. Consider the subgroup Tc(t) of

T/(Fr — id)T = ^/(Fr — idT x %c/(Fr — id)^

generated by Tc/(Fr — id)Tc and the image of t. The isomorphism (iv)^(v) maps Tc (t) into (v)t by the assumption on t. Since (v)t form a compact set, the group Tc(t) must be contained in a compact subset of T/(Fr — id)T. This forces the image of t in Tnc/(Fr — id)Tnc to be trivial. (Indeed, the latter quotient is isomorphic as a topological group to a quotient of Rdim T modulo an R-subspace via the exponential map. So any subgroup generated by a nontrivial element is not contained in a compact set.) Therefore t can be chosen in Tc.

It remains to verify that (iv)t, (iv)t and (v)t are in bijection. Clearly (iv) ^ (iv)t is onto. As we have just seen that (iv)t ^ (v)t, it suffices to observe that (v)t ^ (iv)t is onto, which is a standard fact [for instance in the context of the (twisted) Weyl integration formula for K x Fr]. □

3.3 Plancherel measure on the unramified spectrum

Lemma 3.2 provides a bijection GA,ur,temp ~ QF\ac, which is in fact a topological isomorphism. The Plancherel measure Tpl'ur on GA,ur is supported on GA,ur,temp. We would like to describe its pullback measure on Ac, to be denoted Tp1'™^™^ Note that Ac is topologically isomorphic to Tc/(Fr—id)Tc.

(This is induced by the natural surjection Tc ^ Ac.) Fix a measure dt on the latter which is a push forward from a Haar measure on Tc.

Proposition 3.3 The measure T0l,ur'temp pulled back to Tc/(Fr — id)Tc is

^i,ur,temp~ ^ det(1 — ad(t x Fr)|Lie (G)/Lie (fFr)) ^

T0 (t) = C •---^-— dt

0 det(1 — q—1ad(t x Fr)|Lie (G)/Lie (TFr))

for some constant C e Cx, depending on the normalization of Haar measures. Here t e Tc is any lift oft. (The right hand side is independent of the choice oft.)

Proof The formula is due to Macdonald [72]. For our purpose, it is more convenient to follow the formulation as in the conjecture of [98, p. 281] (which also discusses the general conjectural formula of the Plancherel measure due to Langlands). By that conjecture (known in the unramified case),

-pl,ur,temp~ w L(1,a 1(t), r) L(1,a(t), r)

(t ) = C • -=--7-=-(

^ L(0,a(t), r) L(0,a—1(t), r)

where C' e Cx is a constant, a(t): T(F) ^ Cx is the character corresponding to t [via (ii) ^ (v) of Lemma 3.1], and r: LT ^ GL(Lie (LU)) is the adjoint representation. Here LU is the L-group of U (viewed in LB). By unraveling the local L-factors, obtain

-pl,ur,temp~ W det(1 — ad (t x Fr)|Lie (G)/Lie (T))

To (t) = C ----^-^^dt. (3.2)

0 det(1 — q—1ad(t x Fr)|Lie(G)/Lie(T))

Finally, observe that det(1 — q —sad(t x Fr) |Lie (T)/Lie (TFr)) is independent of t (and t). Therefore the right hand sides are the same up to constant in (3.2) and the proposition. □

Remark 3.4 Note that the choice of a Haar measure on G (F) determines the measure Kjpl,ur'temp. For example if the Haar measure on G (F) assigns volume 1 to K then GA'ur'temP has total volume 1 with respect to K0l'ur'temp(7) as implied by the Plancherel formula for 1K. Hence the product C • dt.

4 Automorphic L-functions

According to Langlands conjectures, the most general L-functions should be expressible as products of the principal L-functions L(s, n) associated to cuspidal automorphic representations n of GL (d ) over number fields (for

varying d). The analytic properties and functional equation of such L-functions were first established by Godement-Jacquet for general d ^ 1. This involves the Godement-Jacquet integral representation. The other known methods are the Rankin-Selberg integrals, the doubling method and the Langlands-Shahidi method. The purpose of this section is to recall these analytic properties and to set-up notation. More detailed discussions may be found in [32,55,75], [86, §2] and [52, §5].

In this section and some of the later sections we use the following notation.

• F is a number field, i.e. a finite extension of Q.

• G is a connected reductive group over F (not assumed to be quasi-split).

• Z = Z(G) is the center of G.

• VF (resp. V() is the set of all (resp. all finite) places of F.

• S( := Vf\V(.

• Aa isthemaximal F-splitsubtorus in the center of ResF/Q G ,and Aa,( := Ag (R)0.

4.1 Automorphic forms

Let x : Aa,( ^ Cx be a continuous homomorphism. Denote by L2x(G(F) \G(AF)) the space of all functions f on G(AF) which are square-integrable modulo Aa,( and satisfy f (gy z) = x(z) f (y) for all g e G(F), y e G(Ay) and z e Aa,(. There is a spectral decomposition into discrete and continuous parts

LX(G(F)\G(AF)) = Ldiscx ® L2ont,x' Ldisc,x = 0mdisc,x(n) ■ n

where the last sum is a Hilbert direct sum running over the set of all irreducible representations of G(AF) up to isomorphism. Write ARdisc,x(G) for the set of isomorphism classes of all irreducible representations n of G (AF) such that mdisc,x(n) > 0. Any n e ARdisc,x(G) is said to be a discrete automorphic representation of G(AF). If x is trivial (in particular if Aa,( = {1}) then we write mdisc for mdisc,x.

The above definitions allow a modest generalization. Let Xa be a closed subgroup of Z(AF) containing Aa,( and m : Z(AF) n Xa\Xa ^ Cx be a continuous (quasi-)character. Then LM, Ldisc m, mdisc,M etc can be defined analogously. In fact the Arthur-Selberg trace formula applies to this setting. (See [4, Ch 3.1].)

For the rest of Sect. 4 we are concerned with G = GL(d). Take Xa = Z (Af ) so that m is a quasi-character of Z(F)\Z(Af). Note that Aa,( = Z (F()° in this case. We denote by AM(GL(d, F)) the space consisting of automorphic functions onGL(d , F)\GL(d, Af) which satisfy f (zg) = m(z) f (g)

for all z e Z (AF) and g e GL (d, AF) (see Borel and Jacquet [10] for the exact definition and the growth condition). We denote by Acusp,M(GL(d, F)) the subspace of cuspidal functions (i.e. the functions with vanishing period against all nontrivial unipotent subgroups).

An automorphic representation n of GL(d, AF) is by definition an irreducible admissible representation of GL (d, AF) which is a constituent of the regular representation on AM(GL(d, F)). Then m is the central character of n. The subspace Acusp,M(GL(d, F)) decomposes discretely and an irreducible component is a cuspidal automorphic representation. The notion of cuspidal automorphic representations is the same if the space of cuspidal functions in LM(GL(d, F)\GL(d, Af)) is used in the definition in place of Acusp,M (GL(d, F)), cf. [10, §4.6].

When m is unitary we can work with the completed space LM(GL(d, F)\ GL (d, Af )) of square-integrable functions modulo Z (AF) and with unitary automorphic representations. Note that a cuspidal automorphic representation is unitary if and only if its central character is unitary. We recall the Langlands decomposition of LM(GL(d, F)\GL(d, AF)) into the cuspidal, residual and continuous spectra. What will be important in the sequel is the notion of isobaric representations which we review in Sect. 4.3.

In the context of L-functions, the functional equation involves the con-tragredient representation n. An important fact is that the contragredient of a unitary automorphic representation of GL (d, AF) is isomorphic to its complex conjugate.

4.2 Principal L-functions

Let n = ®vnv be a cuspidal automorphic representation of GL(d, AF) with unitary central character. The principal L-function associated to n is denoted

L(s, n)= L(s, n„).

The Euler product is absolutely convergent when ^e s > 1. The completed L-function is denoted A(s, n), the product now running over all places v e VF. For each finite place v e V(, the inverse of the local L-function L(s, nv) is a Dirichlet polynomial in q-s of degree ^ d. Write

L(s, nv) = fl (1 - * (nv)q-s)-1 . i=1

Note that when nv is unramified, ai (nv) is non-zero for all i and corresponds to the eigenvalues of a semisimple conjugacy class in GLd (C) associated to

Uv ,but when Uv is ramified the Langlands parameters are more sophisticated and we allow some (or even all of) of the ai (Uv) to be equal to zero. In this way we have a convenient notation for all local L-factors.

For each archimedean v, the local L-function L(s, Uv) is a product of d Gamma factors

L(s,Uv) = n Tv(s — m(Uv)), (4.1)

where tr(s) := n —s/2T(s/2) and r<c(s) := 2(2n)—sT(s). Note that r<c(s) = rv(s)rv(s +1) by the doubling formula, so when v is complex, L(s, Uv) may as well be expressed as a product of 2d TR factors.

The completed L-function A(s, U) := L(s, U) nv|^ L(s, Uv) has the following analytic properties. It has a meromorphic continuation to the complex plane. It is entire except when d = 1 and U = |.|it for some t e R, in which case L(s, U) = ZF(s + it) is (a shift of) the Dedekind zeta function of the ground field F with simple poles at s = —it and s = 1 — it. It is bounded in vertical strips and satisfies the functional equation

A(s, U) = e(s, U)A(1 — s,U), (4.2)

where e(s, U) is the epsilon factor and n is the contragredient automorphic representation. The epsilon factor has the form

e(s, U) = e(U)q(U)2 —s (4.3)

for some positive integer q (U) e and root number e(U) of modulus one.

Note that q(U) = q(n), e(n) = e(U) and for all v e VF, L(s, nv) = L(s, Uv). For instance this follows from the fact [42] that n is isomorphic to the complex conjugate U (obtained by taking the complex conjugate of all forms in the vector space associated to the representation U).

The conductor q (U) is the product over all finite places v e VF° of the conductor q (Uv) of Uv. Recall that q (Uv) equals one whenever Uv is unramified. It is convenient to introduce as well the conductor of admissible representations at archimedean places. When v is real we let C(Uv) = nd=;(2 + (Uv)|) and when v is complex we let C(Uv) = nf=1(2 + (Uv)|2). Then we let C (U) be the analytic conductor which is the product of all the local conductors

C(U) :=J1 C(Uv) J] q(Uv) = C(Ux)q(U).

v|ro veVp

Note that C(U) ^ 2 always.

There is 0 ^ 6 < 2 such that

^e im(nv) < 6, resp. logqv |a(nv)| < 6 (4.4)

for all archimedean v (resp. finite v) and 1 ^ i ^ d. When nv is unramified we ask for

|SKe m (nv)| < 6, resp. |logqv la (nv)|| < 6. (4.5)

The value 6 = 1 — is admissible by an argument of Serre and Luo-Rudnick-Sarnak based on the analytic properties of the Rankin-Selberg convolution L(s, n x n). Note that for all v, the local L-functions L(s, nv) are entire on ^e s >6 and this contains the central line ^e s =

The generalized Ramanujan conjecture asserts that all nv are tempered (see [88] and the references herein). This is equivalent to having 6 = 0 in the inequalities (4.4) and (4.5). In particular we expect that when nv is unramified, № (nv)| = 1.

4.3 Isobaric sums

We need to consider slightly more general L-functions associated to non-cuspidal automorphic representations on GL(d, AF). These L-functions are products of the L-functions associated to cuspidal representations and studied in the previous Sect. 4.2. Closely related to this construction it is useful to introduce, following Langlands [70], the notion of isobaric sums of automorphic representations. The concept of isobaric representations is natural in the context of L-functions and the Langlands functoriality conjectures.

Let n be an irreducible automorphic representation of GL (d, AF). Then a theorem of Langlands [10] states that there are integers r ^ 1 andd1; ...,dr ^

1 with d = d1 +-----hdr and cuspidal automorphic representations n1;..., nr

of GL(d1; Af), ••• , GL(dr, AF) such that n is a constituent of the induced representation of n 1 ® ■ ■ ■ ® nr (from the Levi subgroup GL (d1)x- ■ ■ x GL (dr) of GL(d)). A cuspidal representation is unitary when its central character is unitary. When all of n j are unitary then n is unitary. But the converse is not true: note that even if n is unitary, the representation n j need not be unitary in general.

We recall the generalized strong multiplicity one theorem of Jacquet and Shalika [54]. Suppose n and n' are irreducible automorphic representations of GL(d, Af ) such that nv is isomorphic to n'v foralmostall v e VF (wesaythat n and n are weakly equivalent) and suppose that n (resp. n') is a constituent of the induced representation of n1 ® ■ ■ ■ ® nr (resp. n1 ® ■ ■ ■ ® n'r 1). Then r = r' and up to permutation the sets of cuspidal representations {n j}

coincide. Note that this generalizes the strong multiplicity one

theorem of Piatetski-Shapiro which corresponds to the case where U and U are cuspidal.

Conversely suppose U;,..., Ur are cuspidal representations of GL(di, Af ), ..., GL (dr, AF). Then from the theory of Eisenstein series there is a unique constituent of the induced representation of U; Ur whose

local components coincide at each place v e VF with the Langlands quotient of the local induced representation [70, §2]. It is denoted U; ffl ••• ffl Ur and called an isobaric representation (automorphic representations which are not isobaric are called anomalous). The above results of Langlands and Jacquet-Shalika may now be summarized by saying that an irreducible automorphic representation of GL(d, AF) is weakly equivalent to a unique isobaric representation.

We now turn to L-functions. The completed L-function associated to an isobaric representation U = U; ffl ••• ffl Ur is by definition

A(s, U) = H A(s, Uj).

All notation from the previous subsection will carry over to A(s, U). Namely we have the local L-factors L(s, Uv), the local Satake parameters ai (Uv) and /xi (Uv), the epsilon factor e(s, U), the root number e (U), the local conductors q(Uv), C(Uv) and the analytic conductor C(U). The Euler product converges absolutely for ^e s large enough.

One important difference concerns the bounds for local Satake parameters. Even if we assume that U has unitary central character the inequalities (4.4) may not hold. We shall therefore require a stronger condition on U.

Proposition 4.1 Let U be an isobaric representation of GL(d, AF). Assume that the archimedean component U^ is tempered. Then the bounds towards Ramanujan are satisfied. Namely there is a positive constant 6 < 2 such that for all 1 ^ i ^ d and all archimedean (resp. non-archimedean) places v,

^e /(Uv) < 9, resp. logqv \a(U„)| < 9. (4.6)

Proof Let U = U1 El- ■ -EH Ur be the isobaric decomposition with U j cuspidal. Then we will show that all U j have unitary central character, which implies Proposition 4.1.

By definition we have that is a Langlands quotient of the induced representation of U1œ ® ■ ■ ■ ® UrSince is tempered, this implies that all U j œ are tempered, and in particular have unitary central character. Then the (global) central character of U j is unitary as well. □

Remark 4.2 In analogy with the local case, an isobaric representation ni ffl ■ ■ ■ ffl nr where all cuspidal representations n j have unitary central character is called "tempered" in [70]. This terminology is fully justified only under the generalized Ramanujan conjecture for GL(d, AF). To avoid confusion we use the adjective "tempered" for n = ®v nv only in the strong sense that the local representations nv are tempered for all v e VF.

Remark 4.3 In the proof of Proposition 4.1 we see the importance of the notion of isobaric representations and Langlands quotients. For instance a discrete series representation of GL(2, R) is a constituent (but not a Langlands quotient) of an induced representation of a non-tempered character of GL(1, R) x GL(1, R).

4.4 An explicit formula

Let n be a unitary cuspidal representation of GL(d, AF). Let pj (n) denote the zeros of A(s, n) counted with multiplicities. These are also the non-trivial zeros of L(s, n). The method of Hadamard and de la Vallée Poussin generalizes from the Riemann zeta function to automorphic L-functions, and implies that 0 < ^e pj (n) < 1 for all j. There is a polynomial p(s) such that p(s)A(s, n) is entire and of order 1 (p(s ) = 1 except when d = 1 and n = \.\lt, in which case we choose p(s) = (s — it)(1 — it — s)).

The Hadamard factorization shows that there are a = a(n) and b = b(n) such that

p(s)A(s, n) = ea+bsn (1 — pr^j) es/pj^

The product is absolutely convergent in compact subsets away from the zeros p j (n). The functional equation implies that

(pj(n)—1) = —№e b(n).

The number of zeros of bounded imaginary part is bounded above uniformly:

|{ j, |3m p j (n)| < 1} I « log C (n).

Changing n into n ® \.\lt we have an analogous uniform estimate for the number of zeros with |Sm pj (n) — T | ^ 1 (in particular this is «n log T).

Let N(T, n) be the number of zeros with pm pj (n)| ^ T. Then the following estimate holds uniformly in T ^ 1 (Weyl's law):

N(T, n) = T(d log (¿7) + log C(n))

+ log C(n) + On (log T).

The error term could be made uniform in n, see [52, §5.3] for more details.4 The main term can be interpreted as the variation of the argument of C(n)s/2L(s, n along certain vertical segments.

We are going to discuss an explicit formula [see (4.8) below] expressing a weighted sum over the zeros of A(s, n) as a contour integral. It is a direct consequence of the functional Eq. (4.2) and Cauchy formula. The explicit formula is traditionally stated using the Dirichlet coefficients of the L-function L(s, n). For our purpose it is more convenient to maintain the Euler product factorization.

Definte Yj(n) by pj(n) = 1 + iYj(n). We know that |Sm Yj(n)| < 2 and under the GRH, Yj (n) e R for all j.

It is convenient to denote by 2 + irj(n) the (eventual) poles of A(s, n) counted with multiplicity. We have seen that poles only occur when n = \.\lt in which case the poles are simple and {rj (n)} = {t + 2, — t — 2}.

The above discussion applies with little change to isobaric representations. If we also assume that n^ is tempered then we have seen in the proof of Proposition 4.1 that n = n1 ffl ■ ■ ■ ffl nr with n unitary cuspidal representations of GL (d,, A) for all 1 ^ i ^ r .In particular the bounds towards Ramanujan apply and |Sm Yj (n) | < 2 for all j.

Let $ be a Paley-Wiener function whose Fourier transform

$(x )e—2n ixydx (4.7)

has compact support. Note that $ may be extended to an entire function on C.

Proposition 4.4 Let n be an isobaric representation of GL(d, A) satisfying the bounds towards Ramanujan (4.4). With notation as above and for a > 2, the following identity holds

4 One should be aware that Theorem 5.8 in [52] does not apply directly to our setting because it is valid under certain further assumptions on n such as /i, (n„) being real for archimedean places v.

£>(Yj (H)) = £ (H)) + 1°ë2?(n)0(O)

2n l_r

[A'/1 \

— (2 + a + ir,n\$(r — ia)

G+a+ir'n)

+ — I 2 + a + ir,n)$(r + i a)

dr. (4.8)

There is an important remark about the explicit formula that we will use frequently. Therefore we insert it here before going into the proof. The line of integration in (4.8) is away from the zeros and poles because a > 1/2. In particular the line of integration cannot be moved to a = 0 directly. But we can do the following which is a natural way to produce the sum over primes. First we replace A(s, n) by its Euler product which is absolutely convergent in the given region (^e s > 1). Then for each of the term we may move the

line of integration to a = 0 because we have seen that L (s, nv) has no pole for ^e s >6. Thus we have

f ™ A'

IA \ 2

+ a + ir,H\$(r — i a)dr

Q+a+><■•n)

£/—> t( 1+ir'n")

x-[-+ ir,Hv $(r )dr. (4.9)

.. „-^ L V 2

The latter expression is convenient to use in practice. The integral in the right-hand side of (4.9) is absolutely convergent because $ is rapidly decreasing and the sum over v e vf is actually finite since the support of $ is compact.5

Proof The first step is to work with the Mellin transform rather than the Fourier transform. Namely we set

(2+,s)

H\ - + is 1 = s), s e C.

Note that H is an entire function which is rapidly decreasing on vertical strips. This justifies all shifting of contours below. We form the integral

r A' ds

-(s,H)H (s ) — . J (2) A 2i n

5 Note however that it is never allowed to switch the sum and integration symbols in (4.9). This is because the L -function is evaluated at the center of the critical strip in which the Euler product does not converge absolutely.

We shift the contour to ^e s = — 1 crossing zeros and eventual poles of ^ inside the critical strip. The sum over the zeros reads

Y^H (p j (n)) = £ $(Yj (n)) jj

and the sum over the poles reads

— ^ $(rj (n)). j

Note that since e(s, n) = e(U)q(n)1—s we have e'

— (s, n) = —log q(n), s e C. e

We obtain as consequence of the functional Eq. (4.2) that

(A ds r A' ds

— (s,n)H (s)— = —(1 — s,n)H (1 — s ) — J(—1) A 2in J(2) A 2i n

if A' ~\ ds

= — / logq(n) + — (s, n)H(1 — s) — . 7(2 A A J 2i n

Now we observe that

ds 1 H (s) — = —0(0) J(2) 2i n 2n

and also

[A' ds 1 f™ / 3A A'

r A' ~ ds 1 f™ / 3A A'

jm n)H(1 — s)2dn = 2n L $ (r + T) a(2 -

1 r ~ ( 3i) A'

= 2nL $(r + 2) A + irM"tr■

Since A(s, n) = A(s, n) this concludes the proof of the proposition by collecting all the terms above. Precisely this yields the formula when a = 3/2, and then we can make a > 1/2 arbitrary by shifting the line of integration. □

We conclude this section with a couple of remarks on symmetries. The first observation is that the functional equation implies that if p is a zero (resp. pole) of A(s, n) then so is 1 — p (reflexion across the central line). Thus the set {Yj (n)} (resp. [rj (n)}) is invariant by the reflexion across the real axis (namely y goes into Y). Note that this is compatible with the GRH which predicts that ^e pj (n) = 2 and yj (n) e R.

Assuming $ is real-valued the explicit formula is an identity between real numbers. Indeed the Schwartz reflection principle gives $ (s) = $ (s) for all s e C. Because of the above remark the sum over the zeros (resp. poles) in (4.8) is a real; the integrand is real-valued as well for all r e (—<x>, to).

The situation when n is self-dual occurs often in practice. The zeros Yj (n) satisfy another symmetry which is the reflexion across the origin. Assuming n is cuspidal and non-trivial there is no pole. The explicit formula (4.8) simplifies and may be written

z * y m == +n z £ L (2+rHv)

L ' . -

v *(r )dr.

5 Sato-Tate equidistribution

Let G be a connected reductive group over a number field F as in the previous section. The choice of aGal(F/F)-invariant splitting datum (B, T, {Xa }ae Av) as in Sect. 2.1 induces a composite map Gal(F/F) ^ Out(G) ^ Aut(G) with open kernel. Let F1 be the unique finite extension of F in F such that

Gal(F/F) ^ Gal(F1/F) ^ Aut(G).

5.1 Definition of the Sato-Tate measure

Set r1 := Gal(F1 /F). Let K be a maximal compact subgroup of G which is r1 -invariant. (It is not hard to see that such a K exists,cf. [2].)Set Tc := m K. (The subscript c stands for "compact" as it was in Sect. 3.3.) Denote by the Weyl group for (K, K).

Let 9 e r1. Define Qc,e to be the subset of 6 -invariant elements of Qc. Consider the topological quotient K6 of K x 6 by the KT-conjugacy equivalence relation. Set Tc,6 := Tc/(6 — id)Tc. Note that the action of Qc,6 on Tc induces an action on Tc,6. The inclusion Tc K induces a canonical topological isomorphism (cf. Lemma 3.2)

T - Zj/Qc^- (5.1)

The Haar measure on K (resp. on Tc) with total volume 1 is written as

(resp. /k ). Then on K x 6 induces the quotient measure /k (so that

for any continuous function f1 on K6 and its pullback f on K, J f ^j/K — J f jf) thus also a measure / f ^ on Tc,6.

Definition 5.1 The 6-Sato-Tate measure KST on Tc,6/Qc,6 is the measure

transported from /k via (5.1). K6

Lemma 5.2 Let KST) denote the measure on Tc,6 pulled back from KST on

Tc,6/Qc,6 [so that J f KST) = I f KST for every continuous f on Tc,6/Qc,6 and its pullback f ]. Then

where D6(t) = det(1 — ad(t x 6)\Lie (K)/Lie (TC)) and t signifies a parameter on Tc,6.

Proof The twisted Weyl integration formula tells us that for a continuous

f: K ^ C,

if (k)/k =77^ L D6(t )i ^f (x—1tx6) ■dxdt. Jk \^c,6 \j Kg JKt6 \ k

Notice that ft6 is the twisted centralizer group of t in K (or, the centralizer group of 16 in K). On the right hand side, / f 6 is used for integration. When

f is a pullback from K6, the formula simplifies as

I f (k)/K = 777" : L De(t)f (t) ■ jfce JKe 6 \Qc,e \Jfrj lc'°

and the left hand side is equal to Jf f (t^ST by definition. □

5.2 Limit of the Plancherel measure versus the Sato-Tate measure

Let 6, t e r1. Then clearly Qc,6 = Qct6t—1, K6 — Kt6t—1 via k ^ x(k) and Tc,6 — Tct6t—1 via t ^ T(t). Accordingly KST and KST are identified with Kttt—1 and Kttt—1 0, respectively.

Fix once and for all a set of representatives C(T1) for conjugacy classes in r1. For 6 e C(r1), denote by [6] its conjugacy class. For each finite place v

such that G is unramified over Fv, the geometric Frobenius Frv e Gal(FVr/Fv) gives a well-defined conjugacy class [Frv] in T. The set of all finite places v of F where G is unramified is partitioned into

{Vf (d)h eC (Ti)

such that v e Vf(9) if and only if [Frv] = [9].

For each finite place v of F, the unitary dual of G(Fv) and its Plancherel measure are written as GA and /xpl. Similarly adapt the notation of Sect. 3.1 by appending the subscript v.Now fix 9 e C(T1) and suppose that G is unramified at v and that v e VF(9). We choose F ^ Fv such that Frv has image 9 in T1 (rather than some other conjugate). This rigidifies the identification in the second map below. (IfFrv maps to r9r-1 then the second map is twisted by t.)

a ,„• tomn canonical ^ ^

G (Fv) r p ~ TCFrv/^c,Frv = Tc,e/^c,e- (5.2)

By abuse of notation let Tv 'ur ' temp [a measure on G(Fv)A'ur ' temp] also denote the transported measure on Tc^/ i2 c,o. Let Cv denote the constant of Proposition 3.3, which we normalize such that Tv1'0ur'temp has total volume 1. Note that TST also has total volume 1.

Proposition 5.3 Fix any 9 e C(r1). As v ^ to in VF(9), we have weak

^pl,ur,temp ^ST

convergence \\v ^ \ST as v ^to.

Proof It is enough to show that TfO11''1^ ^ TST on Tc,e as v tends to to in VF(9). Consider the measure Tv1'1ur'temp := C-1^1)™''1^. It is clear from the formula of Proposition 3.3 that Tv1'1ur'temp ^ TST as v ^to in VF (9). In particular, the total volume of Tv1'1ur,temp tends to 1, hence Cv ^ 1as v ^ to in VF(9). We conclude that T^O^'1^ ^ TST) as desired. □

Remark 5.4 The above proposition was already noticed by Sarnak for G = SL(n) in [90, §4].

5.3 The generalized Sato-Tate problem

Let n be a cuspidal6 tempered automorphic representation of G (AF) satisfying

Hypothesis. The conjectural global L-parameter for n has Zariski dense image in LGF1/F.

6 If n is not cuspidal then the hypothesis is never supposed to be satisfied.

Of course this hypothesis is more philosophical than practical. The global Langlands correspondence between (L-packets of) automorphic representations and global L-parameters of G (AF) is far from established. A fundamental problem here is that global L-parameters cannot be defined unless the conjectural global Langlands group is defined. (Some substitutes have been proposed by Arthur in the case of classical groups. The basic idea is that a cuspidal automorphic representation of GLn can be put in place of an irreducible n-dimensional representation of the global Langlands group.) Nevertheless, the above hypothesis can often be replaced with another condition, which should be equivalent but can be stated without reference to conjectural objects. For instance, when n corresponds to a Hilbert modular form of weight ^ 2 at all infinite places, one can use the hypothesis that it is not a CM form (i.e. not an automorphic induction from a Hecke character over a CM field).

Let us state a general form of the Sato-Tate conjecture. Let qv denote the cardinality of the residue field cardinality at a finite place v of F. Define Vf(6, := {v e Vf(6, n) : qv < x} for x e

Conjecture 5.5 Assume the above hypothesis. For each 9 e C(Ti), let Vf(9,n) be the subset of v e VF(9) such that nv is unramified. Then {nv}veyF(9,n) are equidistributed according to ^ST. More precisely

1 - ^ST

— u9 as x -œ.

|Vf(9,n)^x | <

veVF (9,n)<x

Z ^ - vf

The above conjecture is deemed plausible in that it is essentially a consequence of the Langlands functoriality conjecture at least when G is (an inner form of) a split group. Namely if we knew that the L-function L (s, n, p) for any irreducible representation LG ^ GLd were a cuspidal automorphic L-function for GLd then the desired equidistribution is implied by Theorem 1 of [92, AppA.2].

Remark 5.6 In general when the above hypothesis is dropped, it is likely that n comes from an automorphic representation on a smaller group than G. [If factors through an injective L-morphism LHF1/F ^ LGF1/F then the Langlands functoriality predicts that n arises from an automorphic representation of H(Af).] Suppose that the Zariski closure of Im (y„) in LGF1/F is isomorphic to LHF1/F for some connected reductive group H over F. (In general the Zariski closure may consist of finitely many copies of LHF1/F.) Then {nv}veyF(e,n) should be equidistributed according to the Sato-Tate measure belonging to H in order to be consistent with the functoriality conjecture.

One can also formulate a version of the conjecture where v runs over the set of all finite places where nv are unramified by considering conjugacy classes

in LGF1/f rather than those in G x 9 for a fixed 9. For this let K^ denote the quotient of K by the equivalence relation coming from the conjugation by K x T1. Since K ^ is isomorphic to a suitable quotient of Tc, the Haar measure on K gives rise to a measure, to be denoted TST, on the quotient of Tc. Let VF (n) (where x e ) denote the set of finite places of F such that nv are unramified and qv ^ x .By writing v ^to we mean that qv tends to infinity.

Conjecture 5.7 Assume the above hypothesis. Then as x ^ to the set {nv: v e VF(n)^x} is equidistributed according to TfT. Namely

1 ^ ' - —ST

as x ^ to.

Vf I .

Remark 5.8 Unlike Conjecture 5.5 it is unnecessary to choose embeddings F F v to rigidify (5.2) since the ambiguity in the rigidification is absorbed in the conjugacy classes in LGFl/F. The formulation of Conjecture 5.7 might be more suitable than the previous one in the motivic setting where we would not want to fix F Fv. The interested reader may compare Conjecture 5.7 with the motivic Sato-Tate conjecture of [96, 13.5].

The next subsection will discuss the analogue of Conjecture 5.5 for auto-morphic families. Conjecture 5.7 will not be considered any more in our paper. It is enough to mention that the analogue of the latter conjecture for families of algebraic varieties makes sense and appears to be interesting.

5.4 The Sato-Tate conjecture for families

The Sato-Tate conjecture has been proved for Hilbert modular forms in [6,7]. Analogous equidistribution theorems in the function field setting are due to Deligne and Katz. (See [59, Thm9.2.6] for instance.) Despite these fantastic developments, we have little unconditional theoretical evidence for the Sato-Tate conjecture for general reductive groups over number fields. On the other hand, it has been noticed that the analogue of the Sato-Tate conjecture for families of automorphic representations is more amenable to attack. Indeed there was some success in the case of holomorphic modular forms and Maass forms [34, Thm2] and [53,83,97]. The conjecture has the following coarse form, which should be thought of as a guiding principle rather than a rigorous conjecture. Compare with some precise results in Sect. 9.7.

Heuristic 5.9 Let {Fk }k^1 be a "general" sequence of finite families of auto-morphic representations of G(AF) such that IFk I ^ œ as k ^ <x. Then {nv e Fk} are equidistributed according to pùf1 as k and v tend to infinity

subject to the conditions that v e VF (6) and that all members of Fk are unramified at v.

We are not going to make precise what "general" means, but merely remark that it should be the analogue of the condition that the hypothesis of Sect. 5.3 holds for the "generic fiber" of the family when the family has a geometric meaning (see also [87]). In practice one would verify the conjecture for many interesting families while simply ignoring the word "general". Some relation between k and v holds when taking limit: k needs to grow fast enough compared to v (or more precisely \Fk \ needs to grow fast enough compared to qv).

It is noteworthy that the unpleasant hypothesis of Sect. 5.3 can be avoided for families. Also note that the temperedness assumption is often unnecessary due to the fact that the Plancherel measure is supported on the tempered spectrum. This is an indication that most representations in a family are globally tempered, which we will return to in a subsequent work.

Later we will verify the conjecture for many families in Sect. 9.7 as a corollary to the automorphic Plancherel theorem proved earlier in Sect. 9. Our families arise as the sets of all automorphic representations with increasing level or weight, possibly with prescribed local conditions at finitely many fixed places.

6 Background materials

This section collects background materials in the local and global contexts. Sections 6.1 and 6.3 are concerned with p-adic groups while Sects. 6.4,6.5 and 6.8 are with real and complex Lie groups. The rest is about global reductive groups.

6.1 Orbital integrals and constant terms

We introduce some notation in the p-adic context.

• F is a finite extension of Qp with integer ring O and multiplicative valuation \ ■ \.

• G is a connected reductive group over F.

• A is a maximal F-split torus of G, and put M0 := ZG(A).

• K is a maximal compact subgroup of G corresponding to a special point in the apartment for A.

• P = MN is a parabolic subgroup of G over F, with M and N its Levi subgroup and unipotent radical, such that M d M0.

• y e G (F) is a semisimple element. (The case of a non-semisimple element is not needed in this paper.)

• IY is the neutral component of the centralizer of y in G. Then IY is a connected reductive group over F.

• fG (resp. fIy) is a Haar measure on G(F) (resp. IY(F)).

• is the quotient measure on IY (F)\G(F) induced by fG and fIy .

№Iy y

• 0 e Cc°°(G(F)).

• Dg(y) := n« |1 — a(y)\ for a semisimple y e G(F), where a runs over the set of roots of G (with respect to any maximal torus in the connnected centralizer of y in G) such that a(Y) = 1. Let M be an F-rational Levi subgroup of G. For a semisimple y e G(F), we define DGM(y) similarly by further excluding those a in the set of roots of M.

Define the orbital integral

oG(F)(0, fiG, Illy ) := / 0(x—1Yx) —.

JIy(F )\g( f ) f Iy

When the context is clear, we use OY (0) as a shorthand notation.

We recall the theory of constant terms (cf. [105, p. 236]). Choose Haar measures , fM, , on K, M(F), N(F), respectively, such that fG = fK ffM fN holds with respect to G(F) = KM(F)N(F). Define the (normalized) constant term 0M e M(F)) by

0M(m) = S1J2(m) / / 0(kmnk~i)fKfN. (6.1)

Jn (f )J K

Although the definition of 0M involves not only M but P, the following lemma shows that the orbital integrals of 0M depend only on M by the density of regular semisimple orbital integrals, justifying our notation.

Lemma 6.1 For all (G, M)-regular semisimple y e M(F),

Oy(0M, fM, fIy) = DM(Y)1/2Oy(0, fG, fIy).

Proof [105, Lem 9]. (Although the lemma is stated for regular elements y e G, it suffices to require y to be (G, M)-regular. See Lemma 8 of loc. cit.) □

It is standard that the definition and facts we have recollected above extend to the adelic case. (Use [63, §§7-8], for instance). We will skip rewriting the analogous definition in the adelic setting.

Now we restrict ourselves to the local unramified case. Suppose that G is unramified over F .Let B c P C G be Borel and parabolic subgroups defined over F. Write B = TU and P = MN where T and M are Levi subgroups such that T c M and U and N are unipotent radicals.

Lemma 6.2 Let 0 e Hur(G). Then SM (0) = 0M, in particular SG (0) = SM (SM 0) = 0t.

Proof Straightforward from (2.1) and (6.1). □

6.2 Gross's motives

Now let F be a finite extension of Q (although Gross's theory applies more generally). Let G be a connected reductive group over F and consider its quasi-split inner form G*. Let T* be the centralizer of a maximal F-split torus of G*. Denote by Q the Weyl group for (G*, T*) over F. Set V = Gal(F/F). Gross [47] attaches to G an Artin-Tate motive

MotG = 0 MotG,d (1 - d)

with coefficients in Q. Here (1 — d) denotes the Tate twist. The Artin motive MotG,d (denoted Vd by Gross) may be thought of as a V -representation on a Q-vector space whose dimension is dimMotG,d. Define

L(MotG) := L(0, MotG)

to be the Artin L-value of L (s, MotG) at s = 0. We recall some properties of MotG from Gross's article.

Proposition 6.3 (i) MotG,d is self-dual for each d ^ 1.

(ii) dim MotG,d = rG = rkG.

(iii) Xd^1(2d — 1) dimMotG,d = dim G.

(iv) \Q\ = Ud>\ ddimMotG,d.

(vi) IfT* splits over a finite extension E of F then the V -action on MotG factors through Gal(E/F).

The Artin conductor f(MotG,d) is defined as follows. Let F' be the fixed field of the kernel of the Artin representation Gal( F / F) ^ GL( Vd) associated to MotG,d. For each finite place v of F, let w be any place of F' above v. Let V(v)i := Gal(F'w/Fv)i (i ^ 0) denote the i-th ramification subgroups. Set

f(Gvd):= X VHdm (Vd/vPV)i)■ (6.2)

i eZ^o

which is an integer independent of the choice of w. Write pv for the prime ideal of OF corresponding to v. If v is unramified in E then f (Gv, d) = 0.

Thus the product makes sense in the following definition.

p f (Gv,d)

f(Motc,d) := n Pv

Let E be the splitting field of T* (which is an extension of F) and set sGspl := [E: F].

Lemma 6.4 For every finite place v of F,

f (Gv, d) < (dimMotG,d) ■ (s? (1 + eFv/Qp logp sf^ — 1) .

Proof Let F', w and Vd be as in the preceding paragraph. Then F c F' c E. Set sv := [F'w : Fv] so that sv ^ sj . The case sv = 1 is obvious (in which case f (Gv, d) = 0), so we may assume sv ^ 2. From (6.2) and Corollary 6.9 below,

f (G v, d) dim (Vd / Vdr(v)i) < (dim Vd )(sv(1 + eFv/Qp logp sv) — 1). i ^0

Recall that wG = \ Q \ is the cardinality of the absolute Weyl group. Let sG be the degree of the smallest extension of F over which G becomes split. The following useful lemma implies in particular that s^f ^ wGsG.

Lemma 6.5 [56, Lem 2.2] For any maximal torus T of G defined over F, there exists a finite Galois extension E of F such that [E: F] ^ wGsG and T splits over E.

6.3 Lemmas on ramification

This subsection is meant to provide an ingredient of proof (namely Corollary 6.9) for Lemma 6.4.

Fix a prime p. Let E and F be finite extensions of Q p with uniformizers m E and mF, respectively. Normalize valuations vE : Ex ^ Z and vF : Fx ^ Z such that vE (mE) = vF (mF) = 1. Write eE/F e for the ramification index and DE/F for the different. For a nonzero principal ideal a of OE, we define vE (a) to be vE (a) for any generator a of a. This is well defined.

Lemma 6.6 Let E be a totally ramified Galois extension of F with [E: F] = pn for n ^ 0. Then

ve (De/f) < pn(1 + n ■ eF/Qp) — 1.

Remark 6.7 In fact the inequality is sharp. There are totally ramified extensions E/F for which the above equality holds as shown by Ore. See also [95, §1] for similar results.

Proof The lemma is trivial when n = 0. Next assume n = 1 but allow E/F to be a non-Galois extension. Let f (x) = Xf=0 a/x' e OF[x] (with ap = 1 and vF (a') ^ 1 fori < p) be the Eisenstein polynomial having mE as a root. By [94, III.6, Cor 2], vE(DE/F) = vE(f '(mE)). The latter equals

VE^^Ja'm f-1^ = Mn^ v e (ia/ m f-1) ^ vf (fm|-1) = eE/Qf + f - 1.

This prepares us to tackle the case of arbitrary n. Choose a sequence of subextensions E = F0 d F1 d • •• D Fn = F such that [ Fm: Fm+1] = p (where Fm /Fm+1 may not be a Galois extension). By above, vFm (DFm/Fm+1) ^ eFm/Qf + f - 1 for 0 ^ m ^ n - 1. Hence

n—1 n-1

v E (Df /F ) vE (D Fm / Fm + 1 ) < X fm (eFm/Qf + f - 1)

m=0 m=0

= nfneF/Qf + fn - 1-

Lemma 6.8 Let E be a finite Galois extension of F. Then

vE(Df /F) < [E: F](1 + eF/Qf logf [E: F]) - 1.

Proof Let Et (resp. Eur) be the maximal tame (resp. unramified) extension of F in E. Then vEt (Dft/Eur) = [Et: Eur] - 1 by [94, III.6, Prop 13]. Clearly vEur (DEur/F) = 0. Together with Lemma 6.6, we obtain

ve (D e/f ) = ve (Df/ft) + [E: Et ]v ft (Dft / fur)

< [E: Et] (1 + eft/Qf logf [E: Et])

-1 + [E : Et]([Et : Eur] - 1) = [E: Eur](1 + eF/Qf logf [E: Et]) - 1

< [E: F](1 + eF/Qf logf [E: F]) - 1.

Corollary 6.9 Let E be a finite Galois extension of F. Then the i th ramification grouf Gal( E / F )i is trivial for i = [E: F ](1 + eF/Qf logf [ E: F ]) - 1.

Proof In the notation of section IV.1 of [94], we have Gal(E/F)m = 1 by definition if m = max1=seGal(E/F) iG (s). But the proposition 4 in that section implies that m ^ vE (DE/F), so Lemma 6.8 finishes the proof.

6.4 Stable discrete series characters

In Sects. 6.4 and 6.5 we specialize to the situation of real groups. G is a connected reductive group over R.

AG,g = Ag (R)0 where AG is the maximal split torus in the center of G. Kg is a maximal compact subgroup of G(R) and Kg := Kg AG,g. q (G) := 1 dim« G(R)/Kg e Z».

T is an R-elliptic maximal torus in G. (Assume that such a T exists.) B is a Borel subgroup of G over C containing T. IY denotes the connected centralizer of y e G (R).

(resp. $) is the set of positive (resp. all) roots of T in G over C. Q is the Weyl group for (G, T) over C, and Qc is the compact Weyl group. P : = jUae^ a

% is an irreducible finite dimensional algebraic representation of G (R). X% e X*(T) is the B-dominant highest weight for %. m(%) := minae$+<X% + p, a). We always have m(%) > 0. ndisc(%) is the set of irreducible discrete series representations of G(R) with the same infinitesimal character and the same central character as %. [This is an L-packet for G(R).] • D^(y) := IIa \1 — a(Y)\ for y e T(R), where a runs over elements of $ such that a(y) = 1. [If y is in the center of G(R), Dg(y) = 1.]

If M is a Levi subgroup of G over C containing T, the following are defined in the obvious manner as above: , , QM, PM, Dg. Define QM := [m e Q : a>—l$M c $+}, which is a set of representatives for Q/QM. For each regular y e T(R), let us define (cf. [3, (4.4)])

(Y, %) := (—1)q(G)Dg(y)1/2Dg(Y)—1/2X &n(Y)

n en disc (%)

where is the character function of n. It is known that the function (Y, %) continuously extends to an QM -invariant function on T (R), thus also to a function on M (R) which is invariant under M (R)-conjugation and supported on elliptic elements ([3, Lem 4.2], cf. [45, Lem 4.1]). When M = G, simply (Y,%) = tr %(y).

We would like to have an upper bound for \$M(Y,%) \ that we will need in Sect. 9.5. This is a refinement of [99, Lem 4.8].

Lemma 6.10 (i) dim § = EU$+ ^St•

(ii) There exists a constant c > 0 indefendent of § such that for every elliftic Y e G (R) and §,

| tr §(y)| < c DGG(Y)-1/2

dim § (§) I $+ I - I

Proof Part (i) is the standard Weyl dimension formula. Let us prove (ii). The formula right above the corollary 1.12 in [19] implies that

Itr§(Y)I< DGL(y)-1'2 X x l n + P'

'a' P'y

Note that their M is our IY and that Ia(Y)I = 1 for all a e $ and all elliptic Y e G(R). Hence by (i),

^ < Z nil"tJ,'°'P\ i n K + P,

, , ... «)

< DG(Y^l^ | n) m(*)-(l*+l-l*JYl>. ILe*+ 'a, PIy)

Lemma 6.11 Assume that M is a Levi subgroup of G over R containing an elliptic maximal torus. There exists a constant c0 > 0 independent of % such that for every elliptic y e M (R),

l*m(Y,%)\ / d^(y)-1/2

dim % l - l $+Ml

/ c. \ 'y

Proof As the case M = G is already proved by Lemma 6.10 (ii), we assume that M C G. Fix an elliptic maximal torus T c M. Since every elliptic element has a conjugate in T (R) and both sides of the inequality are conjugate-invariant, it is enough to verify the lemma for y e T (R). In this proof we borrow some notation and facts from [45, pp. 494-498] as well as [3, pp. 272-274]. For the purpose of proving Lemma 6.11, we may restrict to y e r+, corresponding to a closed chamber for the root system of T (R) in G (R). (See page 497 of [45] for the precise definition.) The proof of [45, Lem 4.1] shows that

= Z • tr & Y

where % M is the irreducible representation of M (R) of highest weight m (% + p) — pM. We claim that there is a constant c1 > 0 independent of % such that

\C(M,%)\ < C1

for all m and %. The coefficients c(m,%) can be computed by rewriting the right hand side of [3, (4.8)] as a linear combination of tr %M (y ) using the Weyl character formula. In order to verify the claim, it suffices to point out that

r in A r+Vnit-'e nAtotiAn^ Thic ic nKiiiAiif oe f, ,

'ysX c $ "H

'ysX and RH.

C( Q+sX, R+) in Arthur's (4.8) takes values in a finite set which is independent of % (or t in Arthur's notation). This is obvious: as Q+sX c $v and R+ c $, there are finitely many possibilities for Q+sX and R+

Now by Lemma 6.10 (i), dim f {a'PM> «"«.J,

with C2 = (nae$+(a, P>)(nae$+(a Pm>)-1 > 0. According to Lemma 6.10 (ii), there exists a constant c3 > 0 such that

|tr£(Y)\ . DM(Y)-1/2

dim%M I $M\—\$+m\'

m (%) ly

To conclude the proof, multiply the last two formulas. □

6.5 Euler-Poincaré functions

We continue to use the notation of Sect. 6.4. Let 'fig denote the Euler-Poincaré measure on G(R)/AG,g (so that its induced measure on the compact inner form has volume 1). There exists a unique Haar measure /xg on G(R) which is compatible with and the standard Haar measure on AG,g. Write co% for the central character of % on AG)IX. Let U(co-1) denote the set of irreducible admissible representations of G(R) whose central characters on AG,g are c-1. For n e ïï.(cû-1), define

Xep (n ® :=z (-1)i dim Hi (Lie G (R), K g ® £) . i

Clozel and Delorme [21] constructed abi-K(-finite function e C((G(R)) which transforms under AG,( by and is compactly supported modulo AG,to, such that

Vn e n^-^ , tr n = xep(n ® §).

The following are well-known:

• Xep(n ® §) = 0 unless n e nm-1) has the same infinitesimal character as §v.

• If the highest weight of § is regular then xEP(n ® § ) = 0 if and only if

n e ndiSc(§_v).

• If n e n(m-1) is a discrete series and xEP (n ® §) = 0thenn e ndisc(§ v) and xep (n ® §) = (-1)9 (G). More precisely, dim H' (Lie G (R), K(

§) equals 1 if i = q (G) and 0 if not.

6.6 Canonical measures and Tamagawa measures

We return to the global setting so that F and G are as in Sect. 6.2. Let G( := (ResF/Q G) xqR, to which the contents of Sects. 6.4 and 6.5 apply. In particular we have a measure /x(p on G((R). For each finite place v of F, define /¿l™ := A(MotG (1)) ■ |mGv I in the notation of [47] where |mGv | is the "canonical" Haar measure on G(Fv) as in §11 of that article. When G is unramified over Fv, the measure / assigns volume 1 to a hyperspecial subgroup of G (Fv). In particular,

/can'EP :=n ¿van x /(P

is a well-defined measure on G (AF).

Let /Tama denote the Tamagawa measure on G(F)\G(AF)/AG,so that its volume is the Tamagawa number (cf. [64, p. 629])

t(G) := /Tama(G(F)\G(AF)/AG()

= |n0(Z(G)Gal(F/F>) ■ | ker1 (F, Z(G))|-1. (6.3)

The Tamagawa measure /Tama on G (AF) of [47] is compatible with /Tama if G(F) and Ag,to are equipped with the point-counting measure and the Lebesgue measure, respectively. The ratio of two Haar measures on G (AF) is computed as:

Proposition 6.12 [47, 10.5]

/can'EP _ L(Motg) -IQI/IQcI /Tama = e(G ()2rkR G( '

The following notion will be useful in that the Levi subgroups contributing to the trace formula in Sect. 9 turn out to be the cuspidal ones.

Definition 6.13 We say that G is cuspidal if Go := ResF/QG satisfies the condition that Ac0 xq R is the maximal split torus in the center of G0 xq R.

Assume that G is cuspidal, so that G(R)/ Ag,( contains a maximal R-torus which is anisotropic.

Corollary 6.14

/can,EP(G(F)\G(Af)/Ag() _ t(G) ■ L(Motg) -M/I^cI /EJP(G(F()/AG() = e(G()2[F:Q]rG ■

Proof It suffices to remark that the Euler-Poincare measure on a compact Lie group has total volume 1, hence (G(F()/ Ag() = 1.

6.7 Bounds for Artin L-functions

For later use we estimate the L-value L(MotG) in Corollary 6.14.

Proposition 6.15 Let s ^ 1 and E be a Galois extension of F of degree [ E: F] < s.

(i) For all e > 0 there exists a constant c = c(e, s, F) > 0 which depends only on e, s and F such that the following holds: For all non-trivial irreducible representations p of Gal(E/F),

cd-e < L(1, p) < cd%.

(ii) The same inequalities hold for the residue Ress=1 ZE (s) of the Dedekind zeta function of E.

(iii) There is a constant A1 = A1(s, F) > 0 which depends only on s and F such that for all faithful irreducible representation p of Gal( E / F),

dA)F < NF/Q(fp) < d1^,

where dE/F = Nf/q(De/F) is the relative discriminant of E/F; recall that dE = dF: F]dE/F.

Proof The assertion (ii) is Brauer-Siegel theorem [14, Theorem 2]. We also note the implication (i) ^ (ii) which follows from the formula

Ze (s) = n L (s,P)dim p- (6.4)

where p ranges over all irreducible representations of Gal(E/F).

The proof of assertion (i) is reduced to the 1-dimensional case by Brauer induction as in [14]. In this reduction one uses the fact that if E'/F' is a subextension of E/ F then the absolute discriminant dE/ of E' divides the absolute discriminant dE of E. Also we may assume that E'/F' is cyclic. For a character x of Gal(E'/F') we have the convexity bound L(1, x) ^ cd€E, (Landau). The lower bound for L(1,x) follows from (ii) and the product formula (6.4).

In the assertion (iii) the right inequality follows from the discriminant-conductor formula which implies that fpim(p) \D E / F. The left inequality follows from local considerations. Let v be a finite place of F dividing D E/F; since p is faithful, its restriction to the inertia group above v is non-trivial and therefore v divides fp. Since v(DE/F) is bounded above by a constant A1(s, F) depending only on [E: F] ^ s and F by Lemma 6.8,wehavev(DE/F) ^ A1v(fp) which concludes the proof. □

Corollary 6.16 For all integers R, D, s e and € > 0 there is a constant c1 = c1(€, R, D, s, F) > 0(depending on R, D, s, F and €) with the following property

(i) For any G such that rG ^ R, dim G ^ D, Z(G) is F-anisotropic, and G splits over a Galois extension of F of degree ^ s,

\L(MotG)\ < C1 n NF/Q(f(MotG,d))d— 1 +€-d=1

(ii) There is a constant A20 = A20( R, D, s, F) such that for any G as in (i),

\l(MotG)\ < C1 n qA20■

veRam(G)

The choice A20 = (D+21)Rs max (1 + eFv/Q log s) is admissible.

prime p

Proof The functional equation for Mote reads

L (MotG ) = L (MotG (1))e(MotG )

v^ww , , Lœ (MotG(1))

L ^(MotG )

where e(MotG) = |AfIdG/2Ud>1 NF/Q(f(MotG,d))d-2.

The (possibly reducible) Artin representation for MotG,d factors through Gal(E/F) with [E: F] ^ s by the assumption. Let A1 = A1(s, F) be as in (iii) of Proposition 6.15. For all e > 0, (i) of Proposition 6.15 implies that there is a constant c = c(e, s, F) > 1 depending only on s and F such that

, dim MotG d

IL'" ....... ' " " ' ....... " A

(MotG(1)) I < EI (cNF/®(f(Moto,d))'

< CrGH NF/Q(f(MotG,d))A1rG . d>1

Formula (7.7) of [47], the first equality below, leads to the following bound since only 1 ^ d ^ |_dG2+1 J can contribute in view of Proposition 6.3 (iii).

Lto (MotG (1))

Lto (MotG )

(d 1)!\ dim MotG,d = 2-[F:Q]rG IT / (d 1)! \

-[F :Q]rG

11 V (2n)d J d>1 x v ' '

(LV Jf-

Set C1(R, D, s, F, e) := |Af |d/22-[f:Q]r (LJ!)R. Then we see that

L dG+1 J

|L(MotG)l < C1 n NF/Q(f(MotG,d))d-1 +e d=1

(d -1 +e)-f (Gv,d )

= c1 n n qv

veRam(G) d=1

This concludes the proof of (i).

According to Lemma 6.4, the exponent in the right hand side is bounded by

df(Gv, d) ^ 2 dimMotGd ■ (s (1 + eFv/Qp log s) - 1).

(we have chosen e = 2). The proof of (ii) is concluded by the fact that

y dim Motg.rf = tq ^ R, d >1

see Proposition 6.3 (ii).

Corollary 6.17 Let Q be a connected cuspidal reductive group over F with anisotropic center. Then there exist constants c2 = c2(Q, F) > 0 and A2(Q, F) > 0 depending only on Q and F such that: for any cuspidal F-Levi subgroup M ofQ and any semisimple y e M (F) which is elliptic in M (R),

l (motjm ) < c2 yl qa

veRam( JM )

where JM denote the connected centralizer of y in M. The following choice is admissible:

(do + l)rG wQSo

prime p

Â2 =-2-„max(1 + eFv/Qp log woso)■

Proof According to Lemma 6.5, s!1« ^ wqSg. Apply Corollary 6.16 for each

I^f with R = rG, D = dG and s = wGsG to deduce the first assertion, which obviously implies the last assertion. Note that rkif ^ rG and that dim If < dG.

Instead of using the Brauer-Siegel theorem which is ineffective, we could use the estimates by Zimmert [109] for the size of the regulator of number fields. This yields an effective estimate for the constants c2 and c3 above, at the cost of enlarging the value of the exponents A1 and A2.

6.8 Frobenius-Schur indicator

The Frobenius-Schur indicator is an invariant associated to an irreducible representation. It may take the three values 1, 0, -1. This subsection gathers several well-known facts and recalls some familiar constructions.

The Frobenius-Schur indicator can be constructed in greater generality but the following setting will suffice for our purpose. We will only consider finite dimensional representations on vector spaces over C or R. The representations are continuous (and unitary) from compact Lie groups or algebraic from linear algebraic groups (these are in fact closely related by the classical "unitary trick" of Hurwitz and Weyl).

Let G be a compact Lie group and denote by / the Haar probability measure on G. Let (V, r) be a continuous irreducible representation of G. Denote by x(g) = Tr (r (g)) its character.

Definition 6.18 The Frobenius-Schur indicator of an irreducible representation (V, r) of G is defined by

We have that s(r) e {-1, 0,1} always.

Remark 6.19 More generally if G is an arbitrary group but V is still finite dimensional, then s(r) is defined as the multiplicity of the trivial representation in the virtual representation on Sym2 V - a2 V. This is consistent with the above definition.

Remark 6.20 (i) Let (Vv, rv) be the dual representation of G in the dual Vv.

It is easily seen that s (r) = s (rv). (ii) If G = G1 x G2 and r is the irreducible representation of G on V = V1 ® V2 where (V1, r1) and (V2, r2) are irreducible representations of Gi (resp. G2), then s(r) = s(n)s(r2).

The classical theorem of Frobenius and Schur says that r is a real, complex or quaternionic representation if and only if s (r) = 1, 0 or -1 respectively. We elaborate on that dichotomy in the following three lemma.

Lemma 6.21 (Realrepresentation)Let (V, r) bean irreducible representation of G. The following assertions are equivalent:

(i) s(r) = 1;

(ii) r is self-dual and defined over R in the sense that V ~ V0 ®r C for some irreducible representation on a real vector space V0. (Such an r is said to be a real representation;)

(iii) r has an invariant real structure. Namely there is a G-invariant anti-linear map j : V ^ V which satisfies j2 = 1.

(vi) r is self-dual and any bilinear form on V that realizes the isomorphism

r ~ rv is symmetric; (v) Sym2 V contains the trivial representation (then the multiplicity is exactly

We don't repeat the proof here (see e.g. [93]) and only recall some of the familiar constructions. We have a direct sum decomposition

V ® V = Sym2 V © a2 V.

The character of the representation V ® V is g ^ x(g)2. By Schur lemma the trivial representation occurs in V ® V with multiplicity at most one. In other words the subspace of invariant vectors of Vv ® Vv is at most one. Note that this subspace is identified with HomG (V, Vv) which is also the subspace of invariant bilinear forms on V.

The character of the representation Sym2 V (resp. a2 V) is

2(X(g)2 + X(g2)) resp. 1 (x(g)2 - x(g2))-

From that the equivalence of (i) with (v) follows because the multiplicity of the trivial representation in Sym2 V (resp. a2 V) is the mean of its character. The equivalence of (iv) and (v) is clear because a bilinear form on V is an element of V v ® V v and it is symmetric if and only if it belongs to Sym2 V v.

The equivalence of (ii) and (iii) follows from the fact that j is induced by complex conjugation on V0 ®r C and conversely V0 is the subspace of fixed points by j. Note that a real representation is isomorphic to its complex conjugate representation because j may be viewed equivalently as a G-isomorphism V ^ V. Since V is unitary the complex conjugate representation r is isomorphic to the dual representation rv. In assertion (ii) one may note that the endomorphism ring of V0 is isomorphic to R.

Lemma 6.22 (Complex representation) Let (V, r) be an irreducible representation of G. The following assertions are equivalent:

(i) s(r) = 0;

(ii) r is not self-dual;

(iii) r is not isomorphic to r; (such an r is called a complex representation;)

(iv) V ® V does not contain the trivial representation.

We note that for a complex representation, the restriction ResC/R V (obtained by viewing V as a real vector space) is an irreducible real representation of twice the dimension of V. Its endomorphism ring is isomorphic to C.

Lemma 6.23 (Quaternionic/symplectic representation) Let (V, r ) be an irreducible representation of G. The following assertions are equivalent:

(i) s(r) = -1;

(ii) r is self-dual and cannot be defined over R.

(iii) r has an invariant quaternionic structure. Namely there is a G-invariant

anti-linear map j : V ^ V which satisfies j 2 = -1. (Such an r is called

a quaternionic representation.)

(iv) r is self-dual and the bilinear form on V that realizes the isomorphism r ~ rv is antisymmetric. (Such an r is said to be a symplectic representation;)

(v) 2 V contains the trivial representation (the multiplicity is exactly one).

The equivalence of (iii) and (iv) again comes from the fact that V is uni-tarizable (because G is a compact group). In that context the notion of symplectic representation is identical to the notion of quaternionic representation. Note that for a quaternionic representation, the restriction ResC/R V is an irreducible real representation of twice the dimension of V. Furthermore its ring of endomorphisms is isomorphic to the quaternion algebra H. Indeed the endo-morphism ring contains the (linear) action by i because V is a representation over the complex numbers and together with j and k = ij this is the standard presentation of H.

From the above discussions we see that the Frobenius-Schur indicator can be used to classify irreducible representations over the reals. The endomorphism ring of an irreducible real representation is isomorphic to either R, C or H and we have described a correspondence with associated complex representations.

7 A uniform bound on orbital integrals

This section is devoted to showing an apparently new result on the uniform bound on orbital integrals evaluated at semisimple conjugacy classes and basis elements of unramified Hecke algebras. Our bound is uniform in the finite place v of a number field (over which the group is defined), the "size" of (the support of) the basis element for the unramified Hecke algebra at v as well as the conjugacy class at v.

The main result is Theorem 7.3, which is invoked in Sect. 9.5. The main local input for Theorem 7.3 is Proposition 7.1. The technical heart in the proof of the proposition is postponed to Sect. 7.3, which the reader may want to skip in the first reading. In Appendix B we discuss an alternative approach to Theorem 7.3 via motivic integration.

7.1 The main local result

We begin with a local assertion with a view toward Theorem 7.3 below. Let G be a connected reductive group over a finite extension F of Qp with a maximal F-split torus A. As usual O, m, kF denote the integer ring, a uniformizer and the residue field. Let G be the Chevalley group for G x F F, defined over Z. Let B and T be a Borel subgroup and a maximal torus of G such that B d T. We assume that

• G is unramified over F,

• char kF > wGsG and char kF does not divide the finitely many constants in the Chevalley commutator relations [namely Cij of (7.34)].

(We assume char kF > wGsG to ensure that any maximal torus of G splits over a finite tame extension, cf. Sect. 7.3 below. The latter assumption on char kF depends only on G.) Fix a smooth reductive model over O so that K := G(O) is a hyperspecial subgroup of G(F). Fix a Borel subgroup B of G whose Levi factor is the centralizer of A in G. Denote by v: F x ^ Q the discrete valuation normalized by v(m) = 1 and by DG the Weyl discriminant function, cf. (13.1) below. Set qv := \kF \.

Suppose that there exists a closed embedding of algebraic groups Sspl : G GLm defined over O such that Sspl(T) [resp. Sspl(B)] lies in the group of diagonal (resp. upper triangular) matrices. This assumption will be satisfied by Lemma 2.17 and Proposition 8.1, or alternatively as explained at the start of Sect. 7.4. The assumption may not be strictly necessary but is convenient to have for some later arguments. In the setup of Sect. 7.2 such a Sspl will be chosen globally over Z[1/Q] (i.e. away from a certain finite set of primes), which gives rise to an embedding over O if v does not divide Q.

Proposition 7.1 There exist aG,v, bG,v, eGv ^ 0 (depending on F, G and Sspl) such that

• for every semisimple y e G (F),

• for every X e X*(A) and k e such that ||X|| ^ k,

n ^ n 1-,-G ,,can ,,can\ ^ „aG,v+bG,vK nGs,\-eG v/2 n i\

0 ^ Oy ^x , iig > J ^ qv •D (y) Gv' - (7.1)

Remark 7.2 We chose the notation aGv v etc rather than aG, F etc in anticipating the global setup of the next subsection where F is the completion of a number field at the place v.

Proof For simplicity we will omit the measures chosen to compute orbital integrals when there is no danger of confusion. Let us argue by induction on the semisimple rank rG of G .In the rank zero case, namely when G is a torus, the proposition is true since Oy(tG ) is equal to 0 or 1. Now assume that rG ^ 1 and that the proposition is known for all groups whose semisimple ranks are less than rGs. In the proof we write aG, bG, eG instead of aG,v, bG,v, eG,v for simplicity.

Step 1. Reduce to the case where Z(G) is anisotropic.

Let Ag denote the maximal split torus in Z(G). Set G := G/AG. The goal of

Step 1 is to show that if the proposition for G then it also holds for G. We have

an exact sequence of algebraic groups over O:1 ^ AG ^ G ^ G ^ 1. By taking F -points one obtains an exact sequence of groups

1 ^ Ag(F) ^ G(F) ^ G(F) ^ 1,

where the surjectivity is implied by Hilbert 90 for AG. [In fact G (O) ^ G (O) is surjective since it is surjective on kF-points and G ^ G is smooth, cf. [63, p. 386], but we do not need this.] For any semisimple y e G(F), denote its image in G ( F ) by y. The connected centralizer of y is denoted I y. There is an exact sequence

1 ^ Ag(F) ^ IY(F) ^ 7y(F) ^ 1.

Weseethat G(F) ^ G(F) induces abijection Iy(F)\G(F) — 7y(F)\G(F). Let A be a maximal F-split torus of G, and A be its image in G. For any X e A), denote its image in A) by X. Then

Og( F) (*xg , —in, —r) < f) (rG, —in, —%).

Indeed, this follows from the fact that IY(F)\G(F) — Iy(F)\G(F) c arries

„can tcan _

-in to —cln. As the proposition is assumed to hold for G, the right hand side

—IY —Iy

is bounded by qT+biK ■ DG(y)-ei/2 = qli+biK ■ DG(y)-g/2. Hence the proposition holds for G if we set aG = a-Q, bG = b-Q and eG = e-Q. This finishes Step 1.

Step 2. When Z(G) is anisotropic.

The problem will be divided into three cases depending on y .In each case we find a sufficient condition on aG, bG and eG for (7.1) to be true.

Step 2-1. When y e Z (G )( F ).

In this case the proposition holds for any aG, bG, eG ^ 0 since Oy(txg) = 0 or 1 and Dg(y) = 1.

Step 2-2. When y is non-central and non-elliptic.

Then there exists a nontrivial split torus S c Z(IY). Set M := ZG(S), which is an F -rational Levi subgroup of G. Then IY c M C G. Note that y is (G, M)-regular. Lemma 6.1 reads

0g(f)(1kx(^)k ) = Dgm (Y)-1/2 OM(F\(1k x(m)K )m )■ (7.2)

By conjugation we may assume without loss of generality that X(m) e M ( F ). (To justify, find x e G(F) such that xMx-1 contains A. Then X(m) e

xM(F)x 1 and OM = O•) Moreover by conjugating X we may assume that X is B n M-dominant. We can write

(1KX(m)K)M = Z CX,/1 KM/(m)KM ■ (7.3)

The ordering in the sum is relative to B n M. For any m = /(m), cX/ is equal to

(1k X(m)K) m (m) = 8 P (m )1/2 Ik X(m)K (mn)dn

JN (F)

= qVPP'/]/Gn(mN(F)K n KX(m)K).

Lemma2.13 and the easy inequality {pP, /) ^ {p, X) allow us to deduce that

0 < CX/ < q{vpP'/}/on(KX(m)K) < qdvG +rG +2{p'X)■ The sum in (7.3) runs over the set of

aa • a with aa e — Z, aa ^ 0

. i 8g

such that / e (X*(T)R)+. Here we need to explain 8G: If / X then X — / is a linear combination of positive coroots with nonnegative rational coefficients. The denominators of such coefficients under the constraint cX// = 0 are uniformly bounded, where the bound depends on the coroot datum. We write 8G for this bound.

The above condition on / and || X H ^ k imply that aa ^ k. We get, by using the induction hypothesis for OM,

0 ^ oM(F)((1KX(m)K )M) ^ Z CX/0m (1KM/(m)KM )

< Z CX,qvM+bMk • DM(Y)—eM/2

< (8g(k + 1))lA+lqdG +rG +2{p'X)qaM +bMk • DM(y)—'m/2

^ qdG +rG(8gk +8g+11)+2{p,X)+aM +bMkdM(Y)—eM/2

cg := dG + rG(8g + 1) + 2{p, X) < dG + rG(8g + 1) + !$+|k.

In view of (7.2) it suffices to find aG, bG, eG ^ 0 such that

Dm (y)-X/2 DM (y)-eM/2qaM +CG+(bM +rG $G )k ^ DG (y)-eG/2qaG +bG k

or equivalently

Dm(Y)eG2~1 DM(y)eG-M ^ qaG-aM-cg+(Pg-bM-m$g)k (7 4) whenever a conjugate of y lies in KX(m)K. For each a e

v(1 - a(y)) ^ 0 if v(a(y)) ^ 0,

v(1 — a(y)) = v(a(y)) ^ —b^K if v(a(y)) < 0

where bs is the constant B5 (depending only on G and S and not on v) of Lemma 2.18. Hence

DM(y) = V 1 - a(y)\v < q^MbsK/2

a(Y)=1

and likewise DM(y) ^ qv"m 1 bsk/2. (We divide the exponents by 2 because it cannot happen simultaneously that v(a(Y)) < 0 and v(a— (y)) < 0.) Therefore condition (7.4) on aG, bG, eG is implied by the two conditions

eG ^ max(1, eM), (7.6)

eG - 1 \ \ bsK eG - eM \ \ bSK

2 2 2 2

^ aG - aM - (dG + rG(sg + 1) + \ \ k) + (bG - bM - tgsg)k.

There are only finitely many Levi subgroups M (up to conjugation) giving rise to the triples (aM, bM, eM). It is elementary to observe that (7.7) holds as long as aG and bG are sufficiently large while eG has any fixed value such that (7.6) holds. We will impose another condition on aG, bG, eG in Step 2-3.

Step 2-3. When y is noncentral and elliptic in G.

This case is essentially going to be worked out in Sect. 7.3. Let Z1, Z2 ^ 0 be as in Lemma 7.9 below. By (7.11) and Corollary 7.11 below, (7.1) will hold if

qrvG(dg+1)qlv+Z1Kdg(y)-Z2 < qavG +bgKDG(y)-eG'2. (7.8)

We have DG(y) < ql®lbsk/2 thanks to (7.5) (cf. Step 2-2). So (7.8) (is not equivalent to but) is implied by the combination of the following two inequalities:

—Z2 + y ^ 0. (7.9)

rG (dG + 1) + 1 + Z1K + |$|bH 2 ( — Z 2 + y) < aG + bG K. (7.10)

The latter two will hold true, for instance, if eG has any fixed value greater than or equal to 2Z2 and if aG and bG are sufficiently large. (We will see in Sect. 7.3 below that Z1 and Z2 are independent of X, y and k.)

Now that we are done with analyzing three different cases, we finish Step 2. For this we use the induction on semisimple ranks (to ensure the existence of aM, bM and eM in Step 2-2) to find aG, bG, eG ^ 0 which satisfy the conditions described at the ends of Step 2-2 and Step 2-3. We are done with the proof of Proposition 7.1.

7.2 A global consequence

Here we switch to a global setup. Let F be a number field. For a finite place v of F, let k(v) denote the residue field and put qv := !k(v) |.

• G is a connected reductive group over F.

• Ram(G) is the set of finite places v of F such that G is ramified at Fv.

• G is the Chevalley group for G xF F, and B, T are as in Sect. 7.1.

• Sspl: G GLm, fixed once and for all, is a closed embedding defined over Z[1/R] for alarge enough integer R such that Sspl(T) [resp. Sspl(B)] lies in the group of diagonal (resp. upper triangular) matrices of GLm. The choice of R depends only on G and Sspl. (We defer to Sect. 7.4 more details and the explanation that there exists such a Sspl.)

• Sbad is the set of finite places v such that either v e Ram(G), char k(v) ^ wGsG, char k (v) divides R, or char k (v) divides at least one of the constants for the Chevalley commutator relations for G, cf. (7.34) below.

Examining the dependence of various constants in Proposition 7.1 leads to the following main result of this section. For each finite place v e Sbad, denote by Av a maximal Fv-split torus of G xF Fv.

Theorem 7.3 There exist aG, bG ^ 0 and eG ^ 1 (depending on F, G and Sspl) such that

• for every finite v e Sbad,

• for every semisimple y e G (Fv),

• for every X e X*(Av) and k e such that ||X|| ^ k,

0 < O'G(Fv) (t°, ^ ^v) < qavG +bGk • DG(y)-eG/2.

Remark 7.4 It is worth drawing a comparison between the above theorem and Theorem 13.1 proved by Kottwitz. In the latter the test function (in the full Hecke algebra) and the base p-adic field are fixed whereas the main point of the former is to allow the test function (in the unramified Hecke algebra) and the place v to vary. The two theorems are complementary to each other and will play a crucial role in the proof of Theorem 9.19.

Remark 7.5 In an informal communication Kottwitz and Ngo pointed out that there might be yet another approach based on a geometric argument involving affine Springer fibers, as in [46, §15], which might lead to a streamlined and conceptual proof, as well as optimized values of the constants aG and bG. Appendix B provides an important step in that direction, see Theorem 14.7 which implies that the constants are transferable from finite characteristic to characteristic zero.

Proof Since the case of tori is clear, we may assume that rG ^ 1. Let 6 e C(H). (Recall the definition of r and C(ri) from Sects. 5.1 and 5.2.) Our strategy is to find aG,6, bG,6, eG,6 ^ 0 which satisfy the requirements (7.7),

(7.9), and (7.10) on aG,v, bG,v, eG,v at all v e VF(6)\Sbad. As for (7.7), we inductively find aM,6, bMy6, eM,6 ^ 0 for all local Levi subgroups M of G as will be explained below.

We would like to explain an inductive choice of aM,6, bM,6, eM,6 ^ 0 for a fixed 6. To do so we ought to clarify what Levi subgroups M of G we consider. Let A denote the set of B-positive simple roots for (G, T). Via an identification G xfF ~ G xZ F we may view A as the set of simple roots for G equipped with an action by r1, cf. [9, §1.3]. Note that Frobv acts as 6 e r1 on A for all v e VF(6)\Sbad. According to [9, §3.2], the 6-stable subsets of A are in bijection with G(Fv)-conjugacy classes of Fv-parabolic subgroups of G. For each v e VF(6)\Sbad, fix a Borel subgroup Bv of G over Fv containing the centralizer Tv of Av in G so that the following are in a canonical bijection with one another.

• 6-stable subsets T of A

• parabolic subgroups Pv of G containing Bv

Denote by PT,v the parabolic subgroup corresponding to T and by MT,v its Levi subgroup containing Tv. Here is an important observation. The constants Z1, Z2 (see Remark 7.10 below) and the inequalities (7.7), (7.9), and

(7.10) to be satisfied by aMT,v, bMT,v, eMT,v depend only on 6 and not on

v e VF(0)\Sbad. (We consider the case where G and M of those inequalities are Mr and a Fv -Levi subgroup of Mr, respectively.) Hence we will write aMr,e, bMr,e, eMr,e ^ 0 for these constants. What we need to do is to define them inductively according to the semisimple rank of M such that (7.7), (7.9), and (7.10) hold true. In particular the desired aG,e, bG,e, eG,e will be obtained and the proof will be finished (by returning to the first paragraph in the current proof).

Now the inductive choice of aMr,e, bMr,e, eMr,e is easy to make once the choice of aMa,e, bMa,e, eMa,e has been made for all i2 c r. Indeed, we may choose eMa,e e to fulfill (7.9) and then choose aMa^e, bMa,e to be large enough to verify (7.7) and (7.10). Notice that Z1, Z2, Z3 of (7.10) (which are constructed in Lemma 7.9 below) depend only on the group-theoretic information of Mr (such as the dimension, rank, affine root data, 8Mr of Mr as well as an embedding of the Chevalley form of Mr into GLd coming from Sspl) but not on v, cf. Remark 7.10.

In view of Theorem 13.1 and other observations in harmonic analysis, a natural question is whether it is possible to achieve eG = 1. This is a deep and difficult question which is of independent interest. It was a pleasant surprise to the authors that the theory of arithmetic motivic integration provides a solution. A precise theorem due to Cluckers, Gordon, and Halupczok is stated in Theorem 14.1 below. It is worth remarking that their method of proof is significantly different from that of this section and also that they make use of Theorem 13.1, the local boundedness theorem. Finally it would be interesting to ask about the analogue in the case of twisted or weighted orbital integrals. Such a result would be useful in the more general situation than the one considered in this paper.

7.3 The noncentral elliptic case

The objective of this subsection is to establish Corollary 7.11, which was used in Step 2-3 of the proof of Proposition 7.1 above. Since the proof is quite complicated let us guide the reader. The basic idea, going back to Langlands, is to interpret the orbital integral OG F ^ (tG) in question as the number of points in the building fixed "up to X" under the action of y . The set of such points, denoted XF(y, X) below, is finite since y is elliptic. Then it is shown that every point of XF (y, X) is within a certain distance from a certain apartment, after enlarging the ground field F to a finite extension. We exploit this to bound XF (y, X) by a ball of an explicit radius in the building. By counting the number of points in the ball (which is of course much more tractable than counting | XF (y, X) |) we arrive at the desired bound on the orbital integral. The proof presented here is inspired by the beautiful exposition of [66, §§3-5] but

uses brute force and crude bounds at several places. We defer some technical lemmas and their proofs to Sect. 7.4 below and refer to them in this subsection but there is no circular logic since no results of this subsection are used in Sect. 7.4.

Throughout this subsection the notation of Sect. 7.1 is adopted and y is assumed to be noncentral and elliptic in G (F). (However y need not be regular.) We assume Z(G) to be anisotropic over F as we did in Step 2 of the proof of Proposition 7.1. Then Iy(F) is a compact group, on which the Euler-Poincare measure ¡EJ assigns total volume 1. Our aim is to bound

OG {F)(1k j(w)K, iG, jf"). It follows from [47, Thm5.5] (for the equality) and Proposition 6.3 that

¡jEEP

..can j Iy

I!d>1 det (1 - Frobv qd-1 |(Motiy4)Iv) IH 1(F, Iy)|

d- A dim Motiy,d

<n (•+qvd-0'

< (1 + qvdimIy +1)/2)rklY < (1 + q^G JG < qrvG(dG+1). (7.11)

Thus we may as well bound O^F\lKJ(w)K, ¡¿n J^p).

Let Ty be an elliptic maximal torus of Iy defined over F containing y . By Lemma 6.5, there exists a Galois extension F'/F with

[F': F] < wgSg (7.12)

such that T y is a split torus over F'. Hence I y and G are split groups over F'. Note that F' is a tame extension of F under the assumption that char kF > wGsG. Let A' be a split maximal torus of G over F' such that A x f F' c A'. Since F'-split maximal tori are conjugate over F', we find

y e G(F') such that A' = yTYy-1

and fix such a y. Write O', w' and v' for the integer ring of F', a uniformizer and the valuation on F' such that v'(w') = 1. With respect to the integral model of G over O at the beginning of Sect. 7.1, we put K' := G (Or). A point of G (F)/ K will be denoted x and any of its lift in G (F) will be denoted x. Let x0 e G (F)/K [resp. x'0 e G (F')/K'] denote the element represented by the trivial coset of K (resp. K'). Then x0 (resp. x'0) may be thought of as a base point of the building B(G(F), K) [resp. B(G(F'), K')] and its stabilizer is identified with K (resp. K'). There exists an injection

B(G(F), K) B(G(F'), K') (7.13)

such that B(G(F), K) is the Gal(F'/F)-fixed points of B(G(F'), K'). (This is the case because F' is tame over F.) The natural injection G(F)/K G (F')/K' coincides with the injection induced by (7.13) on the set of vertices.

Define X' e X*(A') by X' := eF/FX (where eF/F is the ramification index of F' over F) so that X'(m') = X(m) and

\\X'\\ = eF'/f l|X|| < eF'/fk. (7.14)

For (the fixed y and) a semisimple element S e G (F'), set

Xf(y, X) := {X e G(F)/K: x—1yx e KX(m)K}

XF'(S, X') := {x' e G(F')/K': (x')—lSx' e K'X'(m')K'}.

By abuse of notation we write x—1yx e KX(m)K for the condition that x—1yx e KX(m)K for some (thus every) lift x e G(F) of x and similarly for the condition on x'. It is clear that XF(y, X) c XF'(y, X') n (G(F)/K). By (3.4.2) of [66],

oG(F} (lKX(m)k, Xg, XEPP) = !Xf(Y, X)|. (7.15)

Our goal of bounding the orbital integrals on the left hand side can be translated into a problem of bounding | XF (y, X)|.

Let Apt(A'(F')) denote the apartment for A'(F'). Likewise Apt(TY(F)) and Apt(Ty(F')) are given the obvious meanings. We have x0 e Apt(A'(F')). The metrics on B(G(F), K) and B(G(F'), K') are chosen such that (7.13) is an isometry. The metric on B(G(F'), K') is determined by its restriction to Apt(A'(F')), which is in turn pinned down by a (non-canonical choice of) a Weyl-group invariant scalar product on X*(A'), cf. [103, §2.3]. Henceforth we fix the scalar product once and for all. Scaling the scalar product does not change our main results of this subsection.

Remark 7.6 For any other tame extension F" of F and a split maximal torus A" of G over F'', we can find an isomorphism X*(A') and X*(A") over the composite field of F' and F'', well defined up to the Weyl group action. So the scalar product on X*(A") is uniquely determined by that on X*(A'). So we need not choose a scalar product again when considering a different y e G (F).

We define certain length functions. Consider an F'-split maximal torus A" of G (for instance A" = TY or A" = A') and the associated set of roots & = &(G, A") and the set of coroots = &V(G, A"). Let lmax($) denote the largest length of a positive coroot in Note that these are independent

of the choice of A" and completely determined by the previous choice of a Weyl group invariant scalar product on X*(A'). It is harmless to assume that we have chosen the scalar product such that the longest positive coroot in each irreducible system of X*(A') has length lmax($).

Fix a Borel subgroup B' of G over F' containing A' so that y-1 B'y is a Borel subgroup containing Ty . Relative to these Borel subgroups we define the subset of positive roots m+(G, A') and m+(G, Ty). LetmHspl be as in Lemma 7.12 below. In order to bound | XF (y, X)| in (7.15), we control the larger set XF/ (S, Xr) by bounding the distance from its points to the apartment for A'.

Lemma 7.7 Let S e A'(F') andx' e G(F')/K'. Then there exist constants C = C(G, &) > 0, cG > 0, and Y = Y(G) e such that whenever (x0-1Sx' e K'X'(w')K' [i.e. whenever x' e XF,(S, X')],

d(xApt(A'(F'))) < lmaxm • C| A+|- Y|$+|wgsg

x E (|v(1 — «-1(S))| + Y(mGmHspl + mGcG + mHspl)k) ,

aem+(G, A')

where the left hand side denotes the shortest distance from x' to Apt( A'( F')).

Proof of Lemma 7.7 Write x' = anx'0 for some a e A'( F') and n e N (F') using the Iwahori decomposition. As both sides of the above inequality are invariant under multiplication by a, we may assume that a = 1. Let XS e X*(A') be such that S e Xs(w')A'(O'). For each Xo e X*(A')+ recall the definition of nG(X0) from (2.6). Let cG > 0 be a constant depending only on G such that every X0 e X*(A') satisfies the inequality {a, X0) ^ cG^X0y for all a e m+(G, A').

Step 1. Show that S—1n—1Sn e K'Xo(w')K' for some Xo e X*(A')+ such thatnG(X0) ^ (mHspl + cG)eF'/Fk.

By Cartan decomposition there exists a B'-dominant X0 e X*(A') such that S—1n—1Sn e K'X0 (w') K'. The condition on S in the lemma is unraveled as (x0)—1n—1Snx0 e K'X'(w')K'. So

S—1n—1Sn e S—1K'X'(w')K' c (K'X—1(w')K')(K'X'(w')K').

Let w be a Weyl group element for A' in G such that wX—1 is B'-dominant. The fact that K'X0 (w')K' intersects (K'X—l(w')K') (K'X'(w')K') implies [16, Prop4.4.4.(iii)] that

{a, X0) ^ (a, wX—1 + X'j, a e m+(G, A').

We have {a, X') ^ cG||X'||. Note also that

v'(a(S)) e [-mHspiHX'H, mHspiIX'I] (7.16)

by Lemma 7.12 since a conjugate of S belongs to K'X'(m')K'. This implies that

(a, wX= v'(wa-1(S)) < mHspl HX'H.

On the other hand ||X'|| ^ eF/Fk according to (7.14). These inequalities imply the desired bound on nG (X0), which is the maximum of {a, Xo) over a e &+(G, A').

Before entering Step 2, we notify the reader that we are going to use the convention and notation for the Chevalley basis as recalled in Sect. 7.4 below. In particular n e N(F') can be written as [cf. (7.33)]

n = xai (Xai) ■ ■ ■ () (7.17)

for unique X^Xa^+} e F'. Step 2. Show that there exists a constant M\^+\ ^ 0 [explicitly defined in (7.20) below] such that v'(Xai) ^ -M\$>+\ for all 1 < i < |$+|. In our setting we compute

/ 1 \ \®+\ S-1n-1Sn = S-1 ( H xai (-Xai ) J S H Xai (Xai) \' = \®+\ / i=1 / 1 \ \®+\

= ( n xai (-a-\S)Xai )m xai (Xai) \i = \^+\ / i=1 \<S>+\

= n Xai ((1 - a-\S))Xai + (7.18)

where the last equality follows from the repeated use of (7.34) to rearrange the terms. Here Pai is a polynomial (which could be zero) in a-!(S) and Xaj with integer coefficients for j < i. It is not hard to observe from (7.34) that Pai has no constant term. As i varies in [1, |$+|], let Y denote the highest degree for the nonzero monomial term appearing in Pai viewed as a polynomial in either a-1 (S) or Xai (but not both).7 Set Y = 1 if Pai = 0. As mentioned above, the

7 For instance if Pai = a-1 (S)2x4; + a-1(S)3X3a. then Y = 4.

positive roots for a given (G, B, T) are ordered once and for all so that Y depends only on G in the sense that for any G having G as its Chevalley form, Y is independent of the local field F over which G is defined.

Applying Corollary 7.14 below, we obtain from (7.18) and the condition S—1n—1Sn e K'X0(w')K' that

v' ((1 — a—1(S))Xai + Pa^j > —mGnGX)- (7.19)

For 1 < i < I m+1, put

Mi := X (yY—j(|v'(1 — a—!(S))| +mGnG(Xo))) j=1 i — 1

+ ^ Yjm HspleF'/Fk. (7.20)

Obviously 0 ^ M1 ^ M2 ^ ••• ^ M|m+|. We claim that for every i > 1,

v'(Xai) ^ —Mi. (7.21)

When i = 1, this follows from (7.19) as Pa1 = 0. (Use the fact that xa1 (a1 Xa1) commutes with any other xaj (ajXaj) in view of (7.34) since a1 is a simple root.) Now by induction, suppose that (7.21) is verified for all j < i. By (7.19),

v'(Xai) + v' (1 — a;—1(S)) ^ min (—mGnGX), v'(Pai)) .

Note that Pai is the sum of monomials of the form a—1 (S)k1 Xa with 1 j j

j, k1, k2 e Z such that 1 ^ j < i and 0 ^ k1, k2 ^ Y. Each monomial satisfies

v' (aj\S)k1 X j = k1v' (a:1 (S)^ + k2v'(Xaj) ^ —YmHspl eF'/f k — Y Mi—1,

where the inequality follows from (7.16), (7.14), the induction hypothesis, and the fact that 0 ^ M j ^ Mi—1. Hence

v'(Pai) ^ — YmHSpleF'/fk — YMi—1.

v'(Xai) ^ min (-mGnGfro), v'(Pai)) - V (l - a-1 (8)} ^ -mGnG(k0) - YmHspieF/Fk - YM,-1

'(1 - a;-1(S)) | = -Mi, as desired. Now that the claim is verified, we have a fortiori

v'(Xa) ^ -M\®+\, V1 < i < (7.22)

For our purpose it suffices to use the following upper bound, which is simpler than M\^+\. Note that we used the upper bound on nG(X0) from Step 1.

< Y\®+i (V(1 - a-1(8)\ + (momHspi + mgCg

+m Hspi )eF '/f (7.23)

Step 3. Find a e A'( F') suchthat a-1 na e K'.

We can choose a sufficiently large C = C(G, S) > 0, depending only on the Chevalley group G and S, and integers aOO e [-C, 0] for a e A+ such that

1 < Z (-a°a)(ß,av) < C, Vß e A

[This is possible because the matrix ({fi, av))paeA+ is nonsingular. For instance one finds a% e Q satisfying the above inequalities for C = 1 and then eliminate denominators in aa by multiplying a large positive integer.] Now put aa := M\®+\aa e [-CM\^+\, 0] and a := XaeA+ aaav(m') e A'(F') so that

< -v(fi(a)) < C • Vfi e A+. (7.24)

In fact (7.24) implies that the left inequality holds for all fi e $+. Hence

a~lna = xai (ai(a)-1 Xa,

e n Ua 'y(Xai )-v(ai (a)) c Y[ Ua Miv+i+v(Xai )■ i=1 i=1

Here we have written Ua,m with m e R for the image under the isomorphism xa : F ~ Ua( F) of the set {a e F : v(a) ^ m}. In light of (7.21), M\q,+ \ + v(Xai) ^ 0. Hence a-1na e K'. Step 4. Conclude the proof.

Step 3 shows that ax0 e Apt(A'(F')) is invariant under the left multiplication action by n on B(G(F'), K'), which acts as an isometry. Recalling that

x — nx o we have

d(x', Apt(A'(F'))) ^ d(nx0, ax'0) — d(nx'0, nax'0) — d(x'0, ax'0)■ (7.25)

On the other hand, for any x' e Apt(A'(F')) and any positive simple coroot av, we have

d(x', av(mT1 x') < lmax(^). (7.26)

Indeed this holds by the definition of lmax($) as the left hand side is the length of av. Since a — IIaeA+ (av (m'))aa with aa e [-CM|$+|, 0], a repeated use of (7.26), together with a triangle inequality, shows that

d(x'o, ax0) < lmax($) ■ C ■ M|$+| ■ | A+|. (7.27)

Lemma 7.7 follows from (7.25), (7.27), (7.22), (7.23), and eF/F < [F' : F] ^ wGsG as we saw in (7.12). □

Since y is elliptic and G is anisotropic over F, Apt(TY(F)) is a singleton. Let x 1 denote its only point. Then the Gal(F' /F)-action on Apt(TY (F')) has x 1 as the unique fixed point. Motivated by Lemma 7.7 we set M(y, k) to be

lmax($)-C |A+| ■ Y |$+| WGSG

x E (|v(1 - «-1(Y))| + Y(mGmHspl + mqCQ + mHspl)k) ae®(G,TY)

and similarly M(S, k) using a e $(G, A') in the sum instead. Note that we are summing over all roots, not just positive roots as in the lemma. This is okay since it will only improve the inequality of the lemma. We do this such that M(y, k) — M(S, k). Indeed the equality is induced by a bijection $(G, TY) ~ $(G, A') coming from any element y' e G(F') such that A' —

y' TY( y') 1 (for example one can take y' = y). Define a closed ball in G (F)/K: for Z e G (F)/K and R ^ 0,

Ball(z, R) := {x e G(F)/K : d(x, z) < R}■

Lemma 7.8 XF(y,X) c Ball (x 1, M(y,k)).

Proof As we noted above, XF (y, X) c XF/ (y, X') = XF/ (y-18y, A'). Lemma 7.7 tells us that

x e Xf(y, X) ^ d(yx, Apt(A'(F'))) < M(S,k) ^ d(x, Apt(Ty(F'))) < M(S,K).

The last implication uses Apt(A'(F')) = yApt(Ty(F')) (recall A' = yTyy-1). We have viewed x as a point of B(G(F'), K') via the isometric embedding B(G(F), K) ^ B(G(F'), K'). In order to prove the lemma, it is enough to check that d(x, x 1) ^ d(x, x2) for every x2 e Apt(Ty(F')). To this end, we suppose that there exists an x2 with

d(x, x 1) > d(x, x2) (7.28)

and will draw a contradiction.

As a e Gal(F'/F) acts on B(G(F'), K') by isometry, d(x,ax2) = d(x, x2). As Apt(Ty(F')) is preserved under the Galois action, ax2 e Apt(Ty(F')). According to the inequality of [103, 2.3], for any x, y, z e B(G(F'), K') and for the unique mid point m = m (x, y) e B(G(F'), K') such that d(x, m) = d(y, m) = 1 d(x, y),

d(x, z)2 + d(y, z)2 ^ 2d(m, z)2 + 2d(x, y)2. (7.29)

Consider the convex hull C of C0 := {ax2}aeGai(FyF). Since C0 is contained in Apt(Ty (F')), so is C. Moreover C0 is fixed under Gal(F'/F), from which it follows that C is also preserved under the same action. [One may argue as follows. Inductively define C+1 to be the set consisting of the mid points m (x, y) for all x, y e C. Then it is not hard to see that C must be preserved under Gal(F'/F) and that U, ^0C- is a dense subset of C.] As C is a compact set, one may choose x3 e C which has the minimal distance to x among the points of C. By construction

d(x3, x) ^ d(x2, x). (7.30)

Applying (7.29) to (x, y, z) = (x3, ax3, x), where a e Gal(F'/F),

_ _ 2 _ _ 2 _ _ 2 _ _ _ 2 1 _ _ 2

2d(x3, x) = d(x3, x) +d(ax3, x) ^2d(m(x3,ax3), x) +-d(x3,ax3) .

As x3,ax3 e C, we also have m(x3,ax3) e C by the convexity of C. The choice of x3 ensures that d (x 3, x) ^ d (m(x3,ax3), x), therefore d(x3,ax3) — 0, i.e. x3 — ax3. Hence x3 is a Gal(F'/F)-fixed point of

Apt(TY (F')). This implies that x3 — x1, but then (7.30) contradicts (7.28).

Lemma 7.9 There exist constants Z1, Z2 ^ 0, independent of y and X, such that

|Ball(x 1, M(y, k))| < qlv+Z1KDg(y)-Z2■

Remark 7.10 A scrutiny into the defining formulas for Z1 and Z2 (as well as Z1 and Z2) at the end of the proof reveals that Z1 and Z2 depend only on the affine root data, the group-theoretic constants for G (and its Chevalley form), and S. An important point is that, in the situation where local data arise from some global reductive group over a number field by localization, the constants Z1 and Z2 do not depend on the residue characteristic p or the p-adic field F as long as the affine root data remain unchanged. This observation is used in the proof of Theorem 7.3 to establish a kind of uniformity when traveling between places in V(6)\Sbad for a fixed 6 e C(T1) in the notation there.

Proof To ease notation we write M for M(y,k) in the proof. Let us introduce some quantities and objects of geometric nature for the building B(G (F), K). Write emax > 0 for the maximum length of the edges of B(G(F), K). For a subset S of B(G(F), K), define Ch+(S) to be the set of chambers C of the building such that C n S contains a vertex. Let v e B(G(F), K) be a vertex. (We are most interested in the case v — x 1.) We put C1(v) to be the union of chambers in Ch+({v}) and define Ci+1 (v) to be the union of chambers in Ch+ (Ci (v)) for all i e so as to obtain a strictly increasing chain {v} C C1 (v) C C2(v) C C3(v) C .... Denote by V; (v) (resp. Chi (v)) the set of vertices (resp. chambers) contained in C; (v) for i e .

Choose any chamber C in B(G(F), K). Define C + to be the union of all chambers in Ch+ (C). Clearly C + is compact and its interior contains the compact subset C. Hence there exists a maximal RG > 0 such that for every point y e C (which may not be a vertex), the ball centered at y of radius RG is contained in C+. Since the isometric action of G (F) is transitive on the set of chambers, RG does not depend on the choice of C. Moreover the ratio lmax($)IRG does not depend on the choice of metric on the building.

From the definitions we have Ball(x 1, RG) c C1(x 1) and deduce recursively that

Ball(x 1, ;Rg) c Vi(J1) c Ci-(x1), Vi e Z^.

Take M' to be the integer such that j ^ m' < Rj + 1 so that in particular

Ball(xi, M) c Vm(X1). (7.31)

Let us bound |Ch1(u)| for every vertex v e B(G(F), K). The stabilizer of v, denoted by Stab(v), acts transitively on Ch1 (v). Let C e Ch1 (v). Then

|Ch1(v)| = |Stab(v)/Stab(C)| < IG(O)/\wl < IG(kF)| < qdG+rg

where Iw denotes an Iwahori subgroup of G(O), which is conjugate to Stab(C). The group Stab(v) may not be hyperspecial, but the first inequality follows from the fact that the hyperspecial has the largest volume among all maximal compact subgroups [103, 3.8.2]. See the proof of Lemma 2.13 for the last inequality.

Each chamber contains dim A +1 vertices as a dim A-dimensional simplex. Hence for each i ^ 1,

|Vi^1)| < (dim A + 1) ■ |Chi^1)|.

On the other hand,

|Chi+l(Xl)| ^ ^ |Ch1(v)| < qdvG +rg |Vi(X^

veVi (X1)

< qdG+rg(dim A + 1) ■|Chi(X 1)|.

We see that Chi(X1) < ql(dG+rg}(dim A + 1)''-1 and thus

Vm(X1) < (dim A + 1)MqM(dG+rg] < (rG + 1)MqM'(dG +rg}.

(7.32)

Note that

M < 1 + M < 1 + C|A+| ■ Y^Wcsc

x ( Z |v(1 - a-1(y))l + Y(mGmHspl + cg + mHspl)k I,

\ae$ /

which can be rewritten in the form

M' < 1 + +Z1K + Z2 |v(1 - a-1(y))l

Since \v(1 -a(y))\ + \v(1-a-1(y))\ ^ v(1 -a(y)) + v(1 -a-1(y))+2bSK in view of (7.5), we have

qM' < q1+(Z1+bsZ2*Dg(y)-Z2.

Returning to (7.31) and (7.32),

| Ball (xi, M)\ < \VM(xi)| < q(rG+1)M'qM'(dG+rg]

< (ql+(Z 1+2b3Z2)K^G(y)-Z2)dg +2rg+1 .

The proof of Lemma 7.9 is complete once we set Z1 and Z2 as follows, the point being that they

• Z1 := (Z1 + 2bs Z2 )(dG + 2tg + 1),

• Z2 := Z2(dG + 2tg + 1).

Corollary 7.11 |OyG(F)(1km^k, ng, )\<qVg(dg+1)ql+Z1KDG(y)-Z2. Proof Follows from (7.15), Lemmas 7.8 and 7.9.

7.4 Lemmas in the split case

This subsection plays a supporting role for the previous subsections, especially Sect. 7.3. As in Sect. 7.2 let G be a Chevalley group with a Borel subgroup B containing a split maximal torus T, all over Z. Let Sq1 : G ^ GLm be a closed embedding of algebraic groups over Q. Let T denote the diagonal maximal torus of GLm, B the upper triangular Borel subgroup of GLm, and N the unipotent radical of B.

Extend sQp1 to a closed embedding Sspl: G ^ GLm defined over Z[1/R] for some integer R such that Sspl(T) [resp. Sspl(B)] lies in the group of diagonal (resp. upper triangular) matrices of GLm. To see that this is possible, find a maximal Q-split torus T of GLm containing sQpl(T). Choose any Borel subgroup B' over Q containing T. Then there exists g e GLm (Q) such that the inner automorphism Int(g): GLm ^ GLm by y ^ gyg-1 carries (B', T') to (B, T). Then sQp1 and Int(g) extend over Q to over Z[1/R] for some R e Z, namely at the expense of inverting finitely many primes [basically those in the denominators of the functions defining sQp1 and Int(g)].

Now suppose that p is a prime not diving R. Let F be a finite extension of Qp with integer ring O and a uniformizer m. The field F is equipped with a unique discrete valuation vF such that vF (m) = 1. Let X e X*(T). We are interested in assertions which work for F as the residue characteristic p varies.

Lemma 7.12 (resp. Corollary 7.14) below is used in Step 1 (resp. Step 2) of the proof of Lemma 7.7.

Lemma 7.12 There exists mHspi e Z>0 such that for every p, F and X as above and for every semisimple 8 e G(O)X(m)G(O) (and for any choice of Ts containing 8),

Va e vF(a(S)) e [— mHspi||A||, mHspi||A||].

Proof The argument is the same as in the proof of Lemma 2.18. The constant m Sspl corresponds to the constant B5 in that lemma. To see that it is independent of p, F and X, it suffices to examine the argument and see that the constant depends only on G, B, T (and the auxiliary choice of a's as in the proof of Lemma 2.17, which is fixed once and for all).

The unipotent radical of B is denoted N. For F as above, let x0 be the hyperspecial vertex on the building of G(F) corresponding to G(O). As usual put :— (G, T) be the set of positive roots with respect to (B, T).

Let us recall some facts about the Chevalley basis. For each a e $+,let Ua denote the corresponding unipotent subgroup equipped with xa: Ga ~ Ua. Order the elements of as a1,...,a|$+| once and for all such that simple roots appear at the beginning. The multiplication map

mult : Uai x ••• x Ua^+l ^ N, (u i, ...,W|$+|) ^ ui ...W|$+|

is an isomorphism of schemes (but not as group schemes) over Z. This can be deduced from [5, Exp XXII, 5.5.1], which deals with a Borel subgroup of a Chevalley group. In particular (since the ordering on $+ is fixed) any n e N( F ) can be uniquely written as

J = Xai (Yai )---Xa^+l(Ya^+l) (7.33)

for unique Yai e Ga (F) ~ F 's. The Chevalley commutation relation ([20, §III]) has the following form: for all 1 ^ i < j ^ |$+| and all Yai e F 's,

: (Yaj )xaj (Yaj ) — Xaj (Yaj )Xai (Yai ) H Xak (C'j(Y*i)C(Ya

c,d >1 ak—cai +d a j

(7.34)

where Cij are certain integers (depending on G) which we need not know explicitly. It suffices to know that, in the cases of F we are interested in, the

constants Cij are units in O (cf. the assumption in the paragraph preceding Proposition 7.1).

We thank Kottwitz for explaining the proof of the following lemma.

Lemma 7.13 Suppose that the Chevalley group G is semisimple and simply connected. Let Q c X *(T) denote the set of fundamental weights and pv e X *(T) the half sum of all positive coroots. Let X e X *(T) and define n0(X) := max«eQ{«, X). For every prime p, every p-adic field F, and every cocharacter X e X *(T) as above, the following is true: in terms of the decomposition (7.33), each y e G(O)X(m)G(O) n N(F) satisfies the inequality

vf(Yi) > -2no(X){ai,pv), 1 < i < \Q+\.

Proof It suffices to check that

m 2n°(X)pW ym -2n0(X)pv e N(O). (7.35)

[Here we write m2no(X')P v for (pv(m))2n0(X).] Indeed, this implies the desired inequality in the lemma since the decomposition (7.33) is defined over O.

Let us introduce some notation. For each m e Q let V« denote the irreducible representation of G( F) of highest weight m on an F-vector space. Write V« = e^eX*(T) for the weight decomposition. The geometric construction of V« and its weight decomposition by using flag varieties gives us a natural O-integral structures V«(O) in Vm such that V«(O) = e^ex*(t)V«^(O),

where Vm,^(O) := Vm(O) n V«Note that each V« receives an action of

Gm via Gm ^ T G. We may consider a coarser decomposition V« = eieZVm,, where V«,i := e^apv^V«^. For any « e Q and V = V«, set V^i := ej>iVj, V^i(O) := V^i n V(O), and Vi(O) := Vt n V(O). Observe that B(F) preserves the filtration [V^i}ieZ and that N(F) acts trivially on

V^i / V^i+1.

As a preparation, suppose that g e G(O)X(m)G(O) and let us prove that gV«(O) c m-no(X)V«(O) for all « e Q. Since G(O) stabilizes V«(O), the latter condition is true if and only if X(m)V«(O) c m-no(X)V«(O), which holds if and only if

{x, X) ^ —no(X)

for all weights x for Vm by considering the weight decomposition. The above inequality for all weights x is equivalent to that for the lowest weight x for Vm. Since x = w0mm for the longest Weyl element wo, the condition is that {—w0m, X) ^ n0(X) for all m. This is verified by the definition of n0(X) since —w0 preserves the set £2.

Now consider m2n°(X)pV (y — 1)m-2«o(X)pV, where y is as in the lemma. Since m2p acts on Vj as mj, we see from this and the last paragraph that for all o e Q and i e Z,

(m 2n°(X)pV (y — 1)m —2no(X)pV )(va, (O)) — (m 2n°(X)pV (y — 1))(m —ino(X) v0, (O))

C m2n°(X)pV(m—(i+1)no(X)Va<>i+1 (O)) c Vo,i(O).

It follows that m2no(X)pvym —2n°(X)pV also preserves Va,i (O), hence Va(O). Therefore the element belongs to N(O) — N(F) n G(O), concluding the proof of (7.35). □

For an arbitrary Chevalley group G and X e X* (T)+, define a nonnegative integer

nG(X) :— max {a, X). (7.36)

Corollary 7.14 Let G be an arbitrary Chevalley group. For every prime p, every p-adic field F, and every cocharacter X e X*(T), there exists a constant m G > o such that the following is true: each y e G(O)X(m)G(O) n N( F), uniquely decomposed as in (7.33), satisfies the inequality

vf(Yi) ^ —2mGnG(X), 1 < i <

Proof The corollary is immediate from the lemma if G is semisimple and simply connected. Indeed, define n1(X) to be the maximum of {a, X) as a runs over A+, the set of simple roots. Observe that both the sets Q and A+ are bases for X* (T)q. By using the change of basis matrix, it is easy to deduce from Lemma 7.13 that for some constant c > o depending only on G,we have that

VF (Yi) ^ — 2cm(X){ai ,pV)

for all p, F, X, and i. A fortiori the same holds with nG(X) in place of n1 (X). The proof is completed by setting m G :— c maxae$+ {a, pV).

It remains to extend from the simply connected case to the general case. As usual write Gad for the adjoint group of G and Gsc for the simply connected cover of Gad. The pair (B, T) induces the Borel pairs (Bad, Tad) for Gad and (Bsc, Tsc) for Gsc. Write and for the associated sets of roots. Let Nad and Nsc denote the unipotent radicals of Bad and Bsc, respectively. Then the natural maps G ^ Gad and Gsc ^ Gad induce isomorphisms N ~ Nad and Nsc ~ Nad as well as set-theoretic bijections ^ and ^ +j.

In particular the ordering on induces unique orderings on and $+,. With respect to these orderings, the decomposition (7.33) is compatible with the maps G ^ Gad and Gsc ^ Gad. From all this it follows that the corollary for Gsc implies that for Gad, and then for G. □

8 Lemmas on conjugacy classes and level subgroups

This section contains several results which are useful for estimating the geometric side of Arthur's invariant trace formula in the next section.

8.1 Notation and basic setup

Let us introduce some global notation in addition to that at the start of Sect. 4.

- Mo is a minimal F -rational Levi subgroup of G.

- AMo is the maximal split F-torus in the center of Mo.

- Ram(G) :— {v e V? : G is ramified at v}.

- S c V? is a finite subset, often with a partition S — So ]j S1.

- r: lG ^ GLd (C) is an irreducible continuous representation such that r is algebraic.

- S: G ^ GLm is a faithful algebraic representation defined over F (or over OF as explained below)

- For any C-subspace H' c C?(G(Fs)), define

supp H — U supp

where the union is taken over e H.

- qS :— 11 veS qV where qV is the cardinality of the residue field at v. (Convention: qs — 1 if S — 0.)

For each finite place v e Ram(G) of F, fix a special point xV on the building of G once and for all, where xV is required to belong to an apartment corresponding to a maximal FV-split torus AV containing AMo. The stabilizer KV of xV is a good special maximal compact subgroup of G(FV) (good in the senseof [16]). Set KM,V :— KV nM(FV) for each FV-rational Levi subgroup M of G containing AV. Then KMvV is a good special maximal compact subgroup of M (Fv).

It is worth stressing that this article treats a reductive group G without any hypothesis on G being split (or quasi-split). To do so, we would like to carefully choose an integral model of G over OF for convenience and also for clarifying a notion like "level n subgroups". We thank Brian Conrad for explaining us crucial steps in the proof below (especially how to proceed by using the facts from [12]).

Proposition 8.1 The F-group G extends to a group scheme G over OF (thus equipped with an isomorphism G xOF F ~ G) such that

- G xOF OF[Ran1(G) ] is a reductive group scheme (cf. [32]),

- G(Ov) = Kv for all v e Ram(G) (where Kv are chosen above),

- there exists a faithful embedding of algebraic groups S: G — GLm over OF for some m ^ 1.

Remark 8.2 If G is split then Ram(G) is empty and the above proposition is standard in the theory of Chevalley groups.

Proof For any finite place v of F, we will write O(v) for the localization of OF at v (to be distinguished from the completion Ov). As a first step there exists an injective morphism of group schemes SF: G — GLm defined over F for some m ^ 1 ([33, Prop A.2.3]. The scheme-theoretic closure G' of G in GLm' is a smooth affine scheme over Spec OF [1/S] for a finite set S of primes of OF by arguing as in the first paragraph of [32, §2]. We may assume that S D Ram(G). By [32, Prop 3.1.9.(1)], by enlarging S if necessary, we can arrange that G' is reductive. For v e Ram(G) we have fixed special points xv, which give rise to the Bruhat-Tits group schemes G(v) over Ov. Similarly for v e S\Ram(G), let^us choose hyperspecial points xv so that the corresponding group schemes G(v) over Ov are reductive.

According to [12, Prop D.4,p. 147] the obvious functor from the category of affine O(v)-schemes to that of triples (X, X(v), f) where X is an affine F-scheme, X(v) is an affine Ov-scheme and f: X xF Fv ~ %(v) xOv Fv is an equivalence. (The notion of morphisms is obvious in each category.) Thanks to its functorial nature, the same functor defines an equivalence when restricted to group objects in each category. For v e Ram(G), apply this functor to the Bruhat-Tits group scheme G(v) over Ov equipped with G xF Fv ~ G(v) x Ov Fv to obtain a group scheme G(v) over O(v).

An argument analogous to that on page 14 of [12] shows that the obvious functor between the following categories is an equivalence: from the category of finite-type OF-schemes to that of triples (X, {X(v)}veS, {fv}veS) where X is a finite-type OF[1/S]-scheme, X(v) is a finite-type O(v)-scheme and fv: X xOF[1/S] F ~ X(v) xO(v) F is an isomorphism. Again this induces an equivalence when restricted to group objects in each category. In particular, there exists a group scheme G over OF with isomorphisms G xoF Of [1/S] ~ G' and G xoF O(v) ^ G(v) for v e S which are compatible with the isomorphisms between G' and G(v) over F. By construction G satisfies the first two properties of the proposition.

We will be done if SF : G — GLm over F extends to an embedding of group schemes over OF. It is evident from the construction of & that SF extends to S': G — GLm over OF[1/S]. For each v e S, SF extends to S(v) : &(v) — GLm over OV thanks to [17, Prop 1.7.6], which can be defined over O(V) using the first of the above equivalences. Then the second equivalence allows us to glue S' and {S (v)}veS to produce an OF-embedding S: G GLm. □

For each finite place v £ Ram(G), G defines areductive group scheme over Ov, so Kv := G(Ov) is a hyperspecial subgroup of G(Fv). Fix a maximal Fv-split torus Av of G which contains AMo such that the hyperspecial point for Kv belongs to the apartment of Av. For each Levi subgroup M of G whose center is contained in Av, define a hyperspecial subgroup KMv := Kv n M(Fv) of M(Fv). At such a v / Ram(G) define Hur(G(Fv)) (resp. Hur(M(Fv))). The constantterm (Sect. 6.1) ofafunctionin Cc°°(G(Fv)) (resp. C™(M(Fv))) will be taken relative to Kv (resp. KMv). When P = MN is a Levi decomposition, we have Haar measures on Kv, M(Fv) and N(Fv) such that the product measure equals ^v>an on G(Fv) (cf. Sect. 6.1) and the Haar measure on M(Fv) is the canonical measure of Sect. 6.6. In particular when G is unramified at v,

vol(Kv n N(Fv)) — 1 (8.1)

with respect to the measure on N(FV).

Let n be an ideal of OF and v a finite place of F. Let v(n) e Z^o be the integer determined by nOV — mV!("n OV. Define KV (m£) to be the Moy-Prasad subgroup G(FV)xv,s of G(FV) by using Yu's minimal congruent filtration as in [1o8] (which is slightly different from the original definition of Moy and Prasad). Yu has shown that G(Fv)Xv,s — ker(G(Ov) — &(Ov/msv)) in [1o8, Cor 8.8]. Set

Ä'~(n) := J] ker(G(Ov) ^ G(Ov/n)) = ]J Kv (w^)

to be considered the level n-subgroup of G(AS,F).

Fix a maximal torus To of G over F and an R-basis Bo of X*(To)R, which induces a function || ■ ||so: X*(To)R — R^o as in Sect. 2.5. For any other maximal torus T, there is an inner automorphism of G inducing To ~ T, so X*(T)R has an R-basis B induced from Bo, well defined up to the action by Q — Q(G, T). Therefore || ■ ||b,g : X*(T)r — R^o is defined without ambiguity. As it depends only on the initial choice of Bo (and To), let us write || ■ || for || ■ ||b,g when there is no danger of confusion.

Let v be a finite place of G, and Tv a maximal torus of G xF Fv (which may or may not be defined over Fv ).Then ||-|| : X*(Tv)R — R^0 is defined without ambiguity via Tv ~ T0 xyFv by a similar consideration as above. Now assume that G is unramified at v. For any maximal Fv - split torus A c G and a maximal torus T containing A over Fv, the function || ■ ||B0 is well defined on X*(T)R (resp. X*(A)r) and invariant under the Weyl group Q (resp. QF). Hence for every v where G is unramified, the Satake isomorphism allows us to define Hur (G (Fv))^K as well as Hur (M (Fv))^K for every Levi subgroup M of G over Fv .WhenG is unramified at S, we put Hur (G ( Fs ))^k := ®vesHui (G (Fv))^K and define Hur(M(Fs))^K similarly.

For the group GLm with any m ^ 1, we use the diagonal torus and the standard basis to define || ■ ||GLm on the cocharacter groups of maximal tori of GLm (cf. Sect. 2.4). For S: G — GLm introduced above, define

Bs := max ||S(e)||GLm. (8.2)

8.2 z-Extensions

A surjective morphism a : H — G of connected reductive groups over F is said to be a z-extension if the following three conditions are satisfied: Hder is simply connected, ker a c Z (H), and ker a is isomorphic to a finite product nResFi/FGL1 for finite extensions Fi of F. Writing Z := ker a, we often represent such an extension by an exact sequence of F-groups 1 — Z — H — G — 1. By the third condition and Hilbert 90, a: H (F) — G (F) is surjective.

Lemma 8.3 For any G, a z-extension a: H — G exists. Moreover, if G is unramified outside a finite set S, where Sg c S c VF, then H can be chosen to be unramified outside S.

Proof It is shown in [76, Prop 3.1] that a z-extension exists and that if G splits over a finite Galois extension E of F then H can be chosen to split over E. By the assumption on G, it is possible to find such an E which is unramified outside S. Since the preimage of a Borel subgroup of G in H is a Borel subgroup of H, we see that H is quasi-split outside S. □

8.3 Rational conjugacy classes intersecting a small open compact subgroup

Throughout this subsection S = S0 ^ S1 is a finite subset of VF? and it is assumed that S0 D Ram(G). Fix compact subgroups US0 and U? of G(FS00) and G(F R), respectively. Let n be an ideal of OF as before, now assumed to be coprime to S, with absolute norm N(n) e Z^1.

Lemma 8.4 Let US1 :— suppHur(G(FS1 ))^k. There exists cS > o independent of S, k and n (but depending on G, S, USo and UF) such that for all n satisfying

t\tc \ \ BsmK

N(n) ^ cSqSS ,

the following holds: if y e G (F) andx 1y x e KS,F(n)USo US1UF for some x e G (Af ) then y is unipotent.

Proof Let y' — x -1y x. We keep using the embedding S: & — GLm over OF of Proposition 8.1. (For the lemma, an embedding away from the primes in So or dividing n is enough.) At each finite place v e So and v \ n, Lemma 2.17 allows us to find S'V : & — GLm over OV which is GLm (OV)-conjugate to S x qf FV such that S'V sends AV into the diagonal torus of GLm.

Write det(S(Y) - (1 - X)) — Xm + am-1 (y)Xm-1 + ■ ■ -+oo(y), where ai (y) e F for o ^ i ^ m — 1. Our goal is to show that at (y) — o for all i. To this end, assuming ai(y) — o for some fixed i, we will estimate |a,(y)Iv at each place v and draw a contradiction. First consider v e S1. We claim that

v(ai(y)) ^ — BSmk

for every y that is conjugate to an element of suppHur(G(FV))^K. To prove the claim we examine the eigenvalues of S'v(y'), which is conjugate to y. We know y' belongs to suppHur(G(FV))^K, so S'v(y0 e GLm(OV)S,V(^(mV))GLm(OV) for some /x e X*(AV) with ||^,|| ^ k. Then || S'V(x)UGLm ^ Bsk. [A priori this is true for Bsv defined as in (8.2), but BS'V — BS as S'V and S are conjugate.] Let k1, k2 e GLm(OV) be such that S'V(Y') — k1 S'V(x(mV))k2. Lemma 2.15 shows that every eigenvalue X of S'V(/(mV))k2k1 [equivalently of S'v(y')] satisfies v(X) ^ — Bsk. If X — 1, wemusthave v(1 — X) ^ — Bsk. This shows that v(a, (y)) ^ — BSik for any i such that at (y) — o. Hence the claim is true.

At infinity, by the compactness of UF, there exists cS > o such that

Iai (y) If < ce

whenever a conjugate of y e GF belongs to UF.

Now suppose that v is a finite place such that v e S1 and v { n. (This includes v e So.) Then a conjugate of S (y) lies in an open compact subgroup of GLm(FV). Therefore the eigenvalues of S(y) are in OV and

i a; (y)Iv ^ 1-

Finally at v|n,wehave S(x ly x) — 1 e ker(GLm (Ov) — GLm (Ov/mv(n))). Therefore

a (y)iv = a (x-1y x )iv < (iniv)m-i.

Now assume that N(n) ^ csq—BsmK. We assert that ai (y) = 0 for all i. Indeed, if ai (y ) = 0 for some i then the above inequalities imply that

1 = n lai(Y)lv < ( n q-BsmK ) cs n lnim-i < q-BsmKcsN(n)-1 < 1

v \veS1 J vin

which is clearly a contradiction. The proof of lemma is finished. □

8.4 Bounding the number of rational conjugacy classes

We begin with a basic lemma, which is a quantitative version of the fact that Fr is discrete in ArF.

Lemma 8.5 Suppose that {Sv e R>0}veyF satisfies the following: Sv = 1 for all but finitely many v and nv Sv < 2—iSix,i. Let a = (a1,..., ar) e ArF. Consider the following compact neighborhood of a

B(a, 8) := {(x1, ..., xr) e ArF : ixi,v - ai,v iv ^ Sv, Vv, V1 ^ i ^ r}.

Then B(a, S) n Fr has at most one element.

Proof Suppose f = (fi )ri=1,Y = (Yi )]=1 e B(a,S) n Fr. By triangular inequalities,

ifi,v Yi,v iv ^

Sv, v f 2Sv, vi?

for each i . Hence nv ifiv — Yi,v iv < 1.Since fi, Yi e F, the product formula forces f i = Yi . Therefore f = Y . □

The next lemma measures the difference between G(F)-conjugacy and G (AF)-conjugacy.

Lemma 8.6 LetXG (resp. XG) be the set of semisimple G(F)-(resp. G(AF)-)conjugacy classes in G(F). For any [y ] e XG, there exist at most (wGsG)rG+1 elements in XG mapping to [y] under the natural surjection Xg — Xg.

Proof Let [y] e XG be an element defined by a semisimple y e G(F). Denote by XY the preimage of [y] in XG. There is a natural bijection

Xy ^ ker(ker1(F, ¡y) — ker1(F, G)). Since i ker1(F, ¡y)I = i ker1(F, Z([y))i by [62, §4.2], we have iXy i ^

i ker1(F, Z(¡y))l.

Let T be a maximal torus in ¡Y defined over F. Lemma 6.5 tells us that T becomes split over a finite extension E/F such that [E: F] ^ wGsG. Then Gal(F/E) acts trivially on T and Z(¡y). The group ker1(E, Z(¡y)) consists of continuous homomorphisms Gal(F/E) — Z(¡y) which are trivial on all local Galois groups. Hence ker1 (E, Z(¡y)) is trivial. This and the inflation-restriction sequence show that ker1(F, Z(¡y)) is the subset of locally trivial elements in H 1(TE/F, Z(¡y)), where we have written FE/F for Gal(E/F). In particular,

|Xy | < |H\fe/f, Z(ly))l.

Let d := |Gal(E/F)| and denote by [d] the d-torsion subgroup. The long

exact sequence arising from 0 — Z(¡y)[d] — Z(¡y) — d(Z(¡y)) — 0 gives rise to an exact sequence

H\fe/f, Z(r)[d]) — H 1(Fe/f, Z(%)) = H\fe/f, Z(%№] — 0.

Let fid denote the order d cyclic subgroup of Cx. Then Z(¡y)[d] — T[d] ~ (p,d )dim T .Hence

|Xy I < IH1 (FE/F, Z(¡y^d])| < IFE/fI • IZ(Jy)[d]|

< d • (d)dim T < (wgsg)dim T+1.

For the proposition below, we fix a finite subset S0 C Vcontaining Ram(G). Also fix compact subsets USo C G(FSo) and Uœ C G(F^). As usual we will write S for So H Si.

Proposition 8.7 Let k e Let S1 c S0 be a finite subset such that G is unramified at all v e S1. Set USi := suppHui(G(FSi))^k, US'œ := nviÉSUSœ Kv and U := Us0Us1 US'œU(X. Define Yg to be the set of semisimple G (A F )-conjugacy classes of y e G( F ) which meet U. Then there exist constants A3, B3 > 0 such that for all S1 and k as above,

Yg I = O (qA3+B3K)

[In other words, the implicit constant for O (•) is independent ofSi and k .]

Remark 8.8 By combining the proposition with Lemma 8.4 we can deduce the following. Underthe same assumption but with U := KS,TO(n)USo USl Uto we have

YgI = 1 + O BkN(n)-C) . for some constants A, B, C > 0.

Proof Our argument will be a quantitative refinement of the proof of [63, Prop 8.2].

Step I. When Gder is simply connected.

Choose a smooth reductive integral model G over OF [ S^ ] for G and an

embedding of algebraic groups S : G — GLm defined over OF [ S^ ] as in Proposition 8.1. Consider

G(Af) - GLm (AF) - Am (8.3)

where the latter map assigns the coefficients of the characteristic polynomial, and call the composite map S'. Set U' := S'(U). Then | U' n Fm | < to since it is discrete and compact. We would like to estimate the cardinality.

Fix {¿v} such that Sv = 1 for all finite places v and nv Sv < 2-|StoI so that the assumption of Lemma 8.5 is satisfied. We will write Bv(x, r) for the ball with center x and radius r in Fv. Since S is defined over OF [ S^ ], clearly

S(USto) c GLm(6SF'X). Thus

,(USto) c m = Yl Bv(0, 1).

v/SUSX

Set := {0} c (ApTO)m. Similarly as above, S/(Us1 ) c (Of,s1 )m. By the compactness of USo and Uto, there exist finite subsets JSo c FSo and J«j c Fm such that

(Uso)c u m Bv($v, 1)j, s\Uto)c u m Bv($v,Sv)

ps0ejs0 \veso j ptoe/» \vesx

Now we treat the places contained in S1. Let T be a maximal torus of G over F.

Since the image of the composite map Tp — Gp — (GLm )p is contained in a maximal torus of GLm, we can choose g = (gij '^j=1 e GLm (F) such that

g S (T-p)g-1 sits in the diagonal maximal torus T of GLd. Fix the choice of T and g once and for all (independently of Si and k) until the end of Step I. Set

Vmin(g) := min;,j v(gij) and Vmax(g) := max;,j v(gij). There exists B6 > 0 such that for any i e X*(T) with ||i|| ^ k, the element gS(i)g-1 e X * (T) satisfies ||gS(i)g-1|| ^ B6k.LetyS1 = (yv)ves1 e US1.Eachyvhastheform yv = k1i(mv)k2 for some ||i|| ^ k and k1, k2 e G(Ov). Since S(G(Ov)) C GLm(Ov), we see that S(yv) is conjugate to S(i(mv))k' in GLm(Fv) for some k' e GLm(Ov). Applying Lemma 2.15 to (gS(i(mv))g-1)(gk'g-1) with u = gk!g-1 and noting that vmin(u) ^ vmin(g) + vmin(g-1), we conclude that each eigenvalue X of S (yv) satisfies

v(X) ^ -B6K + vmin(g) + vmin(g-1)-

Therefore the coefficients of its characteristic polynomial lie in m-m (B6K+A4) Ov, where we have set A4 := -(vmin(g) + vmin(g-1)) ^ 0. To put things together, we see that

mv~m(B6K+A4)Ov) .

[A fortiori E'(Us1 ) C nves^m^1 ^+A4)Ov holds as well.] The right hand side is equal to the union of nveS1 Bv(fiv, 1),as [ftv }veS1 runsover JS1 =

nveS1 Jv, where Jv is a set of representatives for (rn-m{B6K+A4)Ov/Ov)m. Notice that | JS11 = qS^1 (B6K+A4). Finally, we see that

U' = S'(U) C U B(l,S) lie J

where J = JSo x JS1 x JSx J^. Lemma 8.5 implies that

IU' n Fm I < | J1 = IJso I • IJS11• U^I = o (qm12(B6K+A4^ ,

since I Jso I -I Jro I is a constant independent of k and S1.

For each | e U'n Fm, we claim that there are at most m! semisimple G (F)-conjugacy classes in G (F) which map to | via G (F) ^ GLm (F) ^ F , the map analogous to (8.3). Let us verify the claim. Let T' and T' be maximal tori in G and GLm over F, respectively, such that S (T') c T'. Then the set of semisimple conjugacy classes in G (F) [resp. GLm (F)] is in a natural bijection withT '(F )/Q [resp. T (F )/^GLm ]. The map SIT > : T' ^ T induces a map T'(F )/& ^ T'( F)/QGLm. Each fiber of the latter map has cardinality at most m!, hence the claim follows.

Fix ß e U' n Fm. We also fix y e G(F) suchthat E'(y) = ß. We assume the existence of such a y; otherwise our final bound will only improve. We would like to bound the number of G(AF)-conjugacy classes in G(F) which meet U and G(F)-conjugate to y. Let denote the set of roots over F for any choice of maximal torus TY in G. Define V'(y) to be the set of places v of F such that v / S U Sx and a(y) = 1 and |1 — a(y)\v < 1 for at least one a e . Since TY splits over an extension of Fv of degree at most wGsG (Lemma 6.5), 1 — a(y) belongs to such an extension. Hence the inequality |1 — a(y)\v < 1 implies that

—^ __1_

|1 — a(y)\v < qv wgsg < 2 wgsg .

Put V(y) := V'(y) U S U Sto. Clearly | V(y)| < to. Moreover we claim that |V (y)| = O (1) (bounded independently of y ). Set

Cso := sup m |1 — a(Y)ls011 — a(Y)\sa

Y eUs0

which is finite since USo U^ is compact. Then

1 =nr[|1 — a(Y^v = I n n |1 — a(Y)h

v ae$Y \veV(Y)ae®y

1 |V /(Y)|

^ Cs0 n 2 wGsG ^ Cs02 wGsG

veV '(y)

Thus |V'(y)| = O(1) and also |V(y)| = O(1).

We are ready to bound the number of G(AF)-conjugacy classes in G(F) which meet U and are G(F)-conjugate to y. For any such conjugacy class of Y' e G(F), [63, Prop 7.1] shows that y' is G(Ov)-conjugate to y whenever v e V(y). Hence the number of G(AF)-conjugacy classes of such y' is at most uG|, where uG is the constant of Lemma 8.11 below.

Putting all this together, we conclude that |YG I = O (qSm (Bik+A5)) as S1 and k vary. The lemma is proved in this case. Step II: general case.

Now we drop the assumption that Gder is simply connected. By Lemma 8.3, choose a z-extension

1 — Z — H — G — 1.

Our plan is to argue as on page 391 of [63] with a specific choice of CH and CZ below (denoted CH and CZ by Kottwitz). In order to explain this choice, we need some preparation. If v e S U S^, choose KH,v to be a hyperspecial subgroup of H(Fv) such that a(KH,v) — Kv. (Such a Kh,v exists by the argument of [63, p. 386].) We can find compact sets UH,So c H(FSo) and Uh,<X of H(Fto) suchthata(U h,s0) = Us0 anda(UH,<x) = U^. Moreover, in Lemma 8.9 below we prove the following: □

Claim There exists a constant ß > 0 independent of k and S1 with the following property: for any k e we can choose an open compact subset Uh,s1 c suppHur(H)^ßK such that a(UH,s1) = Us1.

Now choose UZ,S1 to be the kernel of a: UH,S1 ^ US1, which is compact and open in Z(FS1). Then choose a compact set Us such that

Uz,s1 US1 Z(F) = Z(A)1. (This is possible since Z(F)\Z(A)1 is compact.8) Set 1 Z

Uh := ( EI KHv ) Uh,s0 Uhs Uh,to, Uz := Uzs U^1

vsusto

and set UlH := Uh n H(Ap)1, Ulz := Uz n Z(Ap)1. Let Yh be defined as in the statement of Proposition 8.7 (with H and UH replacing G and U). Then page 391 of [63] shows that the natural map YH — Y is a surjection, in particular |Y | ^ |YH |. Since Hder is simply connected, the earlier proof implies that |YH I = O (qfl^K+A5) for some B7, A5 > 0. (To be precise, apply the earlier proof after enlarging UH,S1 to suppHm(Hin the definition of UH. Such a replacement only increases | YH |, so the bound on |YH I remains valid.) The proposition follows.

We have postponed the proof of a claim in the proof of Step II above, which we justify now. Simple as the lemma may seem, we apologize for not having found a simple proof.

Lemma 8.9 Claim 8.4 above is true.

Proof As the claim is concerned with places in S1, which (may vary but) are contained in the set of places where G is unramified (thus quasi-split), we may assume that H and G are quasi-split over F by replacing H and G with their quasi-split inner forms.

8 Choose UZ1 to be any open compact subgroup. Then Uz,s1 U^1 Z(F) has a finite index in

Z(A)1 by compactness. Then enlarge U^1 without breaking compactness such that the equality holds.

Choose a Borel subgroup BH of H, whose image B = a(BH) is a Borel subgroup of G. The maximal torus TH c BH maps to a maximal torus T c B and there is a short exact sequence

1 ^ Z ^ TH A T ^ 1.

The action of Gal(F/F) on X*(TH) factors through a finite quotient. Let T be the quotient of Gal(F/F) which acts faithfully on X*(TH). If v e S0 then G is unramified at v, so the geometric Frobenius at v defines a well-defined conjugacy class, say Cv, in T. Let AH,v (resp. Av) be the maximal split torus in TH (resp. T) over Fv. Then AH,v ^ TH and Av ^ T induce X*(AH,v) — X*(Th )Cv and X*( Av) — X *(T )Cv .We claim that X*(Th ) ^ X* (T) induces a surjective map X*(AH,v) ^ X*(Av).

X*(Th) --X,(Thfv^— X*(Ah,v) — Th(Fv)/Th(Ov)

X*(T)*-X*(T)C^-X*(Av) — T(Fv)/T(Ov)

Indeed, we have an isomorphism X*(AH,v) — TH(Fv)/TH(Ov) via i ^ /x(mv) andsimilarly X*(Av) — T(Fv)/T(Ov).Further,a: TH(Fv) ^ T(Fv) is surjective since H1 (Gal(Fv/Fv), Z(Fv )) is trivial (as Z is an induced torus).

Denote by [T] the finite set of all conjugacy classes in T. For C e [T], choose Z-bases BH,C and BC for X*(TH)C and X*(T)C respectively. [Note that the Z-bases BH for X*(T) and B for X*(TH) are fixed once and for all.] An argument as in the proof of Lemma 2.3 shows that there exist constants c(Bc), c(Bh,c) > 0 such that for all x e X*(Th)R and y e X*(T)R,

Ixibhc > c(Bh,c) • ||xhbh, IyIBc < c(Bc) • ||yu■ (8.4)

Set mc := maxy(minx Ixibhc), where y e X*(T)C varies subject to the condition IyIBc ^ 1 and x e X*(TH)C runs over the preimage of y. (It was shown above that the preimage is nonempty.) Then by construction, for every

y e X*(T)C, there exists an x in the preimage of y such that IxIBhc m c I y Ibc .

Recall that US1 = nveS1 Uv where Uv = Kv^(mv)Kv, the union being taken over i e X*(T*)C such that HiHB ^ k. We have seen that there exists ¡H e X*(Th)Cv mapping to i and IihIbHcCv ^ mcv IiIbCv. By (8.4),

HlHHbH ^ mcvc(Bh,Cv)-C(BcV)HIHB■

Take ¡3 := maxCe[£](mCc(BH,C)-1c(BC)). Clearly ¡3 is independent of S1 and k. Notice that ||^H||bh ^ ¡3\\iA\b ^ ¡k.

For each ^ e X*(T)Cv such that ||^||B ^ k, we can choose apreimage ^H of ^ such that H ||bh ^ ¡K.TakeUH,v to be the union of Kh,v^h(^v)Kh,v for those \iH's. By construction a(UH,v) = Uv. Hence UH,S1 := nveS1 UHvV is the desired open compact subset in the claim of Lemma 8.9. □

Corollary 8.10 In the setting of Proposition 8.7, let YG be the set of all semisimple G(F)-conjugacy (rather than G(AF)-conjugacy) classes whose G(Af)-conjugacy classes intersect U. Then there exist constants A6, B8 > 0 such that |Yg | = O (qB8K+A6) as S1 and k vary.

Proof Immediate from Lemma 8.6 and Proposition 8.7.

The following lemma was used in Step I of the proof of Proposition 8.7 and will be applied again to obtain Corollary 8.12 below.

Lemma 8.11 Assume that Gder is simply connected. For each v e VF and each semisimple y e G(F), letnV,Y be the number of G(Fv)-conjugacy classes in the stable conjugacy class of y in G(Fv). Then there exists a constant uG ^ 1 (depending only on F and G) such that one has the uniform bound nv,Y ^ uG for all v and y.

Proof Put r(v) := Gal(Fv/Fv). It is a standard fact that nv,Y is the cardinality of ker(H 1(Fv, Iy) — H 1(Fv, G)). By [63], H 1(Fv, Iy) is isomorphic to the dual of no( Z (7Y)r (v)). Hence nv,Y < I no (Z (Ty)F(v))I. It suffices to show that a uniform bound for |n0(Z(/Y)r(v))| exists.

By Lemma 6.5, there exists a finite Galois extension E/F with [E: F] ^ wGsG such that IY splits over E. Then Gal(F/F) acts on Z(IY) through Gal( E / F). In particular F (v) acts on Z (IY) through a group of order ^ wGsG. Denote the latter group by F(v)'.

Note that there is a uniform bound on the number of connected components [Z(IY) : Z(TY)0] as v and y vary. Indeed it suffices to observe that there are only finitely many isomorphism classes of root data for IY over F (hence also for IY). This is easily seen from the fact that the roots of IY (for a maximal torus containing y) are exactly the roots a of G such that a(Y) = 1. Write Z(ty)0'f(v) for the F(v)-invariants in Z(/Y)0. Since

[no(Z(ty)f(v)) : no(Z(ty)°'f(v))] < [Z(ty)f(v) : Z(ty)°'f(v)]

< [Z(/,): Z(T,)0],

it is enough to show that |n0(Z(IY)°'F(v)) | is uniformly bounded. Now consider the set of pairs

T = {(A, T): |A| ^ wgsg, dimT ^ tg}

consisting of a C-torus T with an action by a finite group A. Two pairs (A, T) and (AT') are equivalent if there are isomorphisms A ~ A' and T ~ T' such that the group actions are compatible. Note that

(T(v)', Z(Ty)0) e T

and that T depends only on G and F. Clearly \n0(TA) \ depends only on the equivalence class of (A, T) e T. Hence the proof will be complete if T consists of finitely many equivalence classes.

Clearly there are finitely many isomorphism classes for A appearing in T. So we may fix A and prove the finiteness of isomorphism classes of C-tori with A-action. By dualizing, it is enough to show that there are finitely many isomorphism classes of Z[A]-modules whose underlying Z-modules are free of rank at most rG. This is a result of [36, §79]. □

Corollary 8.12 There exists a constant c > 0 (depending only on G) such that for every semisimple y e G (F), |n0 (Z < c. (We do not assume

that Gder is simply connected.)

Proof Suppose that Gder is simply connected. The proof of Lemma 8.11 shows that (Gal(E/F), Z(IY)) e T in the notation there, thus there exists c > 0 such that |n0(Z< c for all semisimple y.

In general, let 1 ^ Z ^ H ^ G ^ 1bea z-extension over F so that Z is a product of induced tori and Hder is simply connected. Since H (F) ^ G (F), we may choose a semisimple yH mapping to y . Let IYH denote the centralizer of yH in H. (Since Hder is simply connected, Iyh is connected.) By the previous argument there exists cH > 0 such that |n0(Z(IYH)r)| < cH for any semisimple yH . The obvious short exact sequence 1 ^ Z ^ IYH ^ IY ^ 1 over F gives rise (Sect. 2.1) to a T-equivariant short exact sequence

1 ^ Z(Ty) ^ Z(Tyh) ^ Z ^ 1,

hence by [62, Cor 2.3],

0 ^ coker (X*(Z(Tyh))t ^ X*(Z)r) ^ no(Zfif) ^ no(Z!)T) ^ n0(ZT) = 0. (8.5)

On the other hand, the inclusions Z ^ IYH ^ H induce T-equivariant maps Z(H) ^ Z(IYH) ^ Z. The map Z(H) ^ Z(IYH) is constructed by [63, 4.2], whereas Z(IYH) ^ Z and Z(H) ^ Z are given by Sect. 2.1. (The distinction comes from the fact that typically Iyh ^ H is not normal.) The three maps are compatible in the obvious sense. By the functoriality of X* (-)r, there is a natural surjection

coker (X*(Z(ff))r ^ X +(Z)r) ^ coker (X*(Z(TYH))r ^ X*(Z)r).

The left hand side is finite because it embeds into the finite group no(Z(G)T), again by [62, Cor 2.3]. Going back to (8.5), we deduce

no (Z (/Y)r)| < | no (Z (Th I ■ I coker (X*( Z (T)f ^ X*(T)r)| < ch ■ |no(Z(G)r)1.

The proof is complete as the far right hand side is independent of y . □

For a cuspidal group and conjugacy classes which are elliptic at infinity, a more precise bound can be obtained by a simpler argument, which would be worth recording here.

Lemma 8.13 Let G be a cuspidal F -group. For any y e G (F) such that Y e G(Ft) is elliptic,

no(Z(TY)r)1 ^ 2rk(G/ag).

Proof Via restriction of scalars, we may assume that F = Q without losing generality. Let us prove the lemma when AG is trivial. By assumption there exists an R-anisotropic torus T in G(R) containing y . Thus T - U(l)rk(G) and T ^ IY over R. The former tells us that Tv(x) - {±1}rk(G} and the latter gives rise to Z(lY)v((X) ^ Tr(t) [63, §4]. Hence the assertion follows from

z(ZTy)t ^ Z(TY)r(t) ^ Tr(t) - {±i}rk(G).

In general when AG is not trivial, consider the exact sequence of Q-groups 1 ^ Ag ^ IY ^ IY/Ag ^ 1, whose dual is the T-equivariant exact sequence of C-groups

i ^ Z(vAG) ^ Z(Ty) ^ zTg ^ i.

Thanks to [62, Cor 2.3], we obtain the following exact sequence:

X *( TG )T ^ no ( Z ( V/AG ^ no (Z Vyf) ^ no( TG )T = 1.

Hence |no(Z(TY)r)\ ^ |no(Z(I^JAg)T)|, and the latter is at most 2rk(G/AG) by the preceding argument. □

9 Automorphic Plancherel density theorem with error bounds

The local components of automorphic representations at a fixed finite set of primes tend to be equidistributed according to the Plancherel measure on the unitary dual, namely the error tends to zero in a family of automorphic representations (cf. Corollary 9.22 below). The main result of this section (Theorems 9.16,9.19) is a bound on this error in terms of the primes in the fixed set as well as the varying parameter (level or weight) in the family. A crucial assumption for us is that the group G is cuspidal (Definition 9.7), which allows the use of a simpler version of the trace formula. For the proof we interpret the problem as bounding certain expressions on the geometric side of the trace formula and apply various technical results from previous sections. One main application is a proof of the Sato-Tate conjecture for families formulated in Sect. 5.4 under suitable conditions on the parameters involved. In turn the result will be applied to the question on low-lying zeros in later sections.

9.1 Sauvageot's density theorem on unitary dual

We reproduce a summary of Sauvageot's result [91] from [99, §2.3] as it can be used to effectively prescribe local conditions in our problem. The reader may refer to either source for more detail.

Let G be a connected reductive group over a number field F. Use v to denote a finite place of F. When M is aLevi subgroup of G over Fv, write (M(Fv)) (resp. * (M(Fv))) for the real (resp. complex) torus whose points parametrize unitary (complex-valued) characters of M (Fv) trivial on any compact subgroup of M (Fv ). The normalized parabolic induction of an admissible representation a of M(Fv) is denoted n-indM(a).

Denote by (G (Fv)A) the space of bounded Tv^-measurable functions fv on G (Fv)A whose support has compact image in the Bernstein center, which is the set of C-points of an (infinite) product of varieties. A measure on G(Fv)A will be thought of as a linear functional on the space F(G(Fv)A) consisting of Zv e Bc(G(Fv)A) such that for every Fv-rational Levi subgroup M of G and every discrete series a of M (Fv),

*u(M(Fv)) ^ C given by x ^ %(n-indM(a ® x))

is a function whose points of discontinuity are contained in a measure zero set. (Here n-ind denotes the normalized parabolic induction.) Now for any finite set S of finite places of F, one can easily extend the above definition

to F(G(Fs)A) so that T(ns) e C makes sense for T e F(G(Fs)A) and nS e G(FS)A. We have a map

C»(G(Fs)) ^ F(G(Fs)A), fa ^ T: ns ^ tr ns(fa),

as follows from Proposition 9.6 below. Harish-Chandra's Plancherel theorem states that

I? (T) = fa (1).

Our notational convention is that TS often signifies an element in the image of the above map whereas TS stands for a general element of F (G (FS )A). Sauvageot's theorem allows us to approximate any T e F(G(FS)A) with elements of Cc°°(G(Fs)).

Proposition 9.1 [91, Thm 7.3] Let Ts e F(G(Fs)A). For any e > o, there exist , e C£°(G(FS)) such that

Tpf (tys) < e and Vns e G(Fs)A, |h(ns) - Ts(ns)l < fs(*s).

Conversely, any fS e Bc(G(Fs)) with the above property belongs to F(G(FS)A).

Remark 9.2 It is crucial that fS e F(G (FS)A) has the set of discontinuity in a measure zero set. Otherwise we could take fS to be the characteristic function on the set of points of G(FS)A which arise as the S-components of some n e ^^disc>x(G) with nonzero Lie algebra cohomology. Note that the latter function typically lies outside F(G(FS)A). The conclusions of Theorems 9.26, 9.27 and Corollary 9.22 are false in general if such an fS is placed at So. Namely in that case TFk,S1 (fS1) is often far from zero but Zp(lS) always vanishes.

From here until the end of this subsection let us suppose that G is unram-ified at S. It will be convenient to introduce F(G(Fs)A,ur) and its subspace F(G(Fs)A,ur>temP) in order to state the Sato-Tate theorem in Sect. 9.7. The former (resp. the latter) consists of fs e F(G(Fs)A) such that the support of fS is contained in G(Fs)A'ur [resp. G(Fs)A'ur'temp]. Denote by F(Tc,0 / Qc,e ) the space of bounded TST-measurable functions on TCye / Qc,e whose points of discontinuity are contained in a ZST-measure zero set. Define F(IIveS Tc,ev/ Qc,ev) in the obvious analogous way. By using the topological Satake isomorphism for tempered spectrum [cf. (5.2)]

nW^cA - G(Fs)A'ur'temp

and extending by zero outside the tempered spectrum, one obtains

•^(n W^A ) - F(G ( Fs )A,ur,temp) ^ f(G ( FS )A,ur). (9.1)

\veS J

Although the first two F(■) above are defined with respect to different measures nveS HSJ and /?, the isomorphism is justified by the fact that the ratio of the two measures is uniformly bounded above and below by positive constants (depending on qS) in view of Proposition 3.3 and Lemma 5.2. Note that the space of continuous functions on nveS Tc,0v/Qc,ev [resp. on G(FS)A'ur,temp] is contained in the first (resp. second) term of (9.1), and the two subspaces correspond under the isomorphism.

Corollary 9.3 Let fS e F(G(FS)A,ur). For any e > 0, there exist (S, ir S e Hur(G(Fs)) such that (i) //$(/) < e and (ii) Vns e G(Fs)A'ur, \ Hs(ns) — fs (ns)| ^ fs (ns).

Proof Let (S, irS e C^°(G (FS)) be the functions associated to fS asinPropo-sition 9.1. Then it is enough to replace (S and irS with their convolution products with the characteristic function on nveS Kv.

The following proposition will be used later in Sect. 9.7. For each v e VF (0), the image of f in F(G(Fv)A'ur) via (9.1) will be denoted f.

Proposition 9.4 Let f e F(fc,o/&c,e) and € > 0. There exists an integer k ^ 1 and for all places v e V F (0), there are bounded functions (v,irv e Hur(G(Fv))^K such that Hv(Tv) ^ e and \ fv(n) — (/^¡(n^ ^ for all

n e G(Fv)A'ur.

Proof This is no more than Corollary 9.3 if we only required (v,fv e Hur (G (Fv)) without the superscript ^ k . So we may disregard finitely many v by considering the subset VF(0)^Q of VF (0) consisting of v such that qv ^ Q for some Q > 0. In view of Proposition 5.3, we may choose Q e Z>0 that

Vv e Vf(0)>q, VHe F(Tc,e/^c,e), 2>HST(|H) < Hpl,urdH|) < 2ZSTdH). (9.2)

Fix any w e VF(d)^Q. Corollary 9.3 allows us to find 0w, Hur(G(Fw)) suchthat

xH W'w) < e/8 and Vnw e G(Fw)A'ur, \Tw(*w) - 4w(*w)\ (*w).

Let k0 e Z^o be such that 0w, f'w e Hur (G (Fw))^K0. Now recall that for every v e VF (0) there is a canonical isomorphism [cf. (2.2), Lemma 3.2] between Hur (G (Fv)) and the space of regular functions in the complex variety T0/Qe. Using the latter as a bridge, we may transport 0w, fw to 0v, fv e Hur (G (Fv)) for every v e V f (0). Clearly fa, fv e Hur (G (Fv))^K0 from the definition of Sect. 2.3. Moreover (9.2) and (9.3) imply that for all v e Vf(0)>q,

Tt(f'v) < e/2 and Vn e G(Fv)A,ur,temp, |%(nv) - fa(nv)\ < f'v(*v).

[Observe that ( f) < 2fST( f) = 2fST( f) < 4fJ( f) < e/2 to justify the first inequality.]

To achieve the latter inequality for non-tempered nv e G (Fv)A,ur, we would like to perturb fv in a way independent of v while not sacrificing the former inequality. Since f(nv) = 0 for such nv, what we need to establish is that \Tv(nv)\ ^ fv(nv) for all non-tempered nv e G(Fv)A,ur. To this end, we use the fact that there is a compact subset K of T0/Q0 such that G(Fv)A,ur is contained in K for every v e VF(0) (cf. [11, Thm XI.3.3]). By using the Weierstrass approximation theorem, we find fw e Hur (G (Fw)) such that

fW(f) < e/8,

Vnw e K\G(Fw)A,ur,temp, \fw(nw)\ + \^w(nw)\ < f(nw), Vnw e G(Fw)A'ur'temp, fw(nw) > 0.

Choose k ^ k0 such that fw e Hur (G (Fw))^K and put fw := fw + fw so that f! ( fw) < e/4 and fw e Hur (G (Fw))^K .For each v e Vf (0)>q ,let fv denote the transport of fw just as fv was the transport of f'w in the preceding paragraph. Then fpl( fv) ^ e and fv e Hur(G(Fv))^K as before. Moreover

Vnv e G (Fv)A,ur,temp, \ fv(nv) - fv(nv)\ < f'v (nv) < fv(nv) and for nv e G(Fv)A'ur\G(Fv)A'ur'temp,

\fv(nv) - fa(nv)\ = \fv(nv)\ ^ fvVv) - \fvv (nv) \ ^ fv(nv),

the last inequality following from fv = f + f". □

Remark 9.5 A more direct approach to (9.3) that wouldn't involve Corollary 9.3 would be to use Weierstrass approximation to find^polynomials 0 and f on tc,0/ Qc,0 of degree ^ k such that ! T - f ^ f and then the isomorphism (9.1) to transport 0 and f at the place v.

We note [91, Lemme 3.5] that for any (v e C^°(G(Fv)) there exists a (v e C°(G(Fv)) such that \Hv(nv)\ < Hv(nv) for all n e G(Fv)A. This statement is elementary, e.g. it follows from the Dixmier-Malliavin decomposition theorem. In fact we have the following stronger result due to Bernstein [8].

Proposition 9.6 (Uniform admissibility theorem) For any (v e C^°(G(Fv)) there exists C > 0 such that |tr n((v)\ ^ C for all n e G(Fv)A.

9.2 Automorphic representations and a counting measure Now consider a string of complex numbers

F = {af(n) e QneAR.diSc,*(G)

such that aF (n ) = 0 for all but finitely many n. We think of F as a multi-set by viewing aF (n ) as multiplicity, or more appropriately as a density function with finite support in F as aF (n) is allowed to be in C. There are obvious meanings when we write n e F and \F\ (we could have written n e supp F for the former):

n e F d^f af(n) = 0, \F\ := ^ aT(n).

In order to explain our working hypothesis, we recall a definition.

Definition 9.7 Let H be a connected reductive group over Q. The maximal Q-split torus in Z(H) is denoted AH. We say H is cuspidal if (H/AH) xq R contains a maximal R-anisotropic torus.

If H is cuspidal then H (R) has discrete series representations. (We remind the reader that discrete series always mean "relative discrete series" for us, i.e. those whose matrix coefficients are square-integrable modulo center.) The converse is true when H is semisimple but not in general. Throughout this section the following will be in effect:

Hypothesis 9.8 ResF/q G is a cuspidal group.

Let S = Soil S1 C VF0 be a nonempty finite subset and H0 e F (G (FS0 )A). (It is allowed that either S0 or S1 is empty.) Let

- (level) US'° be an open compact subset of G(AS'°),

- (weight) % = ®v|0£v be an irreducible algebraic representation of

Go xr C = (ResF/QG) xq C = G xf,v C.

Denote by x : AGo ^ Cx the restriction of the central character for £v. Define

F = F(USo, H, S1,£) by

aF(n) := (-1)q(G)mdisc,x(n) dim(nS°)uS'°/(ns0)

(ns1 )XE-p(no® £) e C. (9.4)

Note that 1Ks (nS1) equals 1 if nS1 is unramified and 0 otherwise, and that XEp(n0 ® £) = 0 unless has the same infinitesimal character as £v. The set of n such that aF(n) = 0 is finite by Harish-Chandra's finite-ness theorem. Let us define measures /fF,S1 and HF Sl associated with F on the unramified unitary dual G(FS1 )A,ur, motivated by the trace formula. Put t(G) := ^can'EP(G(F)Ago\G(Af)). For any function / on G(Fs1 )A'ur which is continuous outside a measure zero set, define

_ ^ Vcan (USo) ^ ^

VF,S1 (fS1) := , Z. aF(n)fS1 (nS1). (9.5)

t'(G) dim £

n eAR-discx^)

The sum is finite because aF is supported on finitely many n. Now the key point is that the right hand side can be identified with the spectral side of Arthur's trace formula with the Euler-Poincare function at infinity as in Sect. 6.5 when H = Hs1 for some $s1 e Hur(G(Fs1 )) ([3, pp. 267-268], cf. proof of [99, Prop 4.1]). So to speak, if we write = $s0$s1 $So,

VF,S1 ($S1) = (-1)q(G)—-f.-

1 1 t'(G) dim £

= q{G) Igeom°$£'Xcan'EP)

( ) t(G) dim £

where /spec (resp. /geom) denotes the spectral (resp. geometric) side Arthur's

the invariant trace formula with respect to the measure vcan'EP. Finally if fS0 has the property that H§0 (fS0) = 0 then put

^pl r \ — 1 ^

Vf,S1 := Vs0 ( JS0) Vf,S1.

Remark 9.9 The measure HF Sl is asymptotically the same as the counting measure

HfS (f1) = T^ Z af (n) f1 (ns1).

\ \ n eARdisc,x (G)

associated with the S1 -components of F (assuming \F\ = 0). More precisely if {Fkis a family of Sect. 9.3 below, then T^st/fFi s is a constant tending to 1 as k by Corollary 9.25.

Example 9.10 Let n e ARdisc,x (G). Suppose that the highest weight of % is regular and that S0 = 0. Then n belongs to F if and only if the following three conditions hold: (nS'™)U = 0, n is unramified at S, and e ndisc(%v). When e ndisc(%v), (9.4) simplifies as

ar(n) = mdiSc,x (n) dim(nS,~)U .

Example 9.11 Let f0 be a characteristic function on some relatively compact TS1-measurable subset US0 c G(FS0)A. Assume that S0 is large enough such that G and all members of F are unramified outside S0. Take US0,TO to be the product of Kv over all finite places v e S0. Then for each n e ARd{sc,x(G),

ar(n) = (-1)q(g)xep® %)mdiSc,x(n) (9.7)

if nS0is unramified, nS0 e Us0 (in which case ar(n) = 0 if moreover XEP(n^ ® %) = 0; otherwise ar(n) = 0). If the highest weight of % is regular, xEP® %) = 0 exactly when e ndisc(%v), in which case (9.7) simplifies as

ar (n) = m disc,x (n)-

Compare this with Example 9.10. (The analogy in the case of modular forms is that n as newforms are counted in the current example whereas old-forms are also counted in Example 9.10.) Finally we observe that since the highest weight of % is regular and e ndisc(% v), the discrete automorphic representation n is automatically cuspidal [107, Thm. 4.3]. In the present example the discrete multiplicity coincides with the cuspidal multiplicity.

Remark 9.12 As the last example shows, the main reason to include S0 is to prescribe local conditions at finitely many places (namely at S0) on automorphic families. For instance one can take f0 = f S0 where 0S0 is a pseudo-coefficient of a supercuspidal representation (or a truncation thereof if the center of G is not anisotropic over FS0). Then it allows us to consider a family of n whose S0-components are a particular supercuspidal representation (or an unramified character twist thereof). By using various f0 [which are in general not equal to fS0 for any 0S0 e C^°(G(FS00))] one obtains great flexibility in prescribing a local condition as well as imposing weighting factors for a family.

9.3 Families of automorphic representations

Continuing from the previous subsection (in particular keeping Hypothesis 9.8) let us introduce two kinds of families {Fk}k^1 which will be studied later on. We will measure the size of % in the following way. Let TTO be a maximal torus of GTO over R. For a B-dominant X e X*(TTO), set m(X) := minae$+ {X, a). For % with B-dominant highest weight X%, define m(%) := m(X%).

Let 0Sq e CC°(G(FS0)). [More generally we will sometimes prescribe a local condition at S0 by fS0 e F(G(FS00)A) rather than 0S0.] In the remainder of Sect. 9 we mostly focus on families in the level or weight aspect, respectively described as the following:

Example 9.13 (Level aspect: varying level, fixed weight) Let nk c OF be a nonzero ideal prime to S for each k ^ 1 such that N(nk) = [OF : nk] tends to to as k —> to. Take

Fk := F (KS,TO(nk ),fS0, S1,%) . Then !Fk! — to as k — to.

Example 9.14 (Weight aspect: fixed level, varying weight) For our study of weight aspect it is always supposed that Z(G) = 1 so that Ag to — 1 and X = 1 in order to eliminate the technical problem with central character when weight varies.9 Let {%k }k^1 be a sequence of irreducible algebraic representations of GTO xr C such that m(%k) — to as k — to. Take

Fk := F (US'TO,TS0, Su%k) -Then !Fk! — to as k — to.

Remark 9.15 Sarnak proposed a definition of families of automorphic representations (or automorphic L-functions) in [87]. The above two examples fit in his definition.

9.4 Level aspect

We are in the setting of Example 9.13. Recall that ResF/QG is assumed to be cuspidal. Fix S: G — GLm as in Proposition 8.1 and let Bs and ch be as in (8.2) and Lemma 8.4. Write Lc(M0) for the set of F-rational cuspidal Levi subgroups of G containing the minimal Levi M0.

9 Without the hypothesis that the center is trivial, one should work with fixed central character and apply the trace formula in such a setting. Then our results and arguments in the weight aspect should remain valid without change.

Theorem 9.16 Fix $S0 e Cf(G (FS0)) and £ .LetS-i C V™ be a subset where G is unramified. Let $s1 e Hur(G(FS1 ))^k be such that \$S1 \ ^ 1 on G(FS1). /f Lc(M0) = {G} (in particular ifG is abelian) then Hfk,s1 (fs1) = H§(fS). Otherwise there exist constants Alv, Blv > 0 and Clv ^ 1 such that

Hfk,S11) - 11) = O (qA/v+BlvKN(n)-Clv) (9.8)

as n, k e S1 and $Sl vary subject to the following conditions:

(i) N(n) ^ c^B^,

(ii) no prime divisors of n are contained in S1.

[The implicit constant in O (•) is independent of n, k, S1 and $S1.]

Remark 9.17 When 1° (fS0) = 0 (9.8) is equivalent to

Hf,s1 (fS1) - Hp1 (fS1) = O BlvKN(n)-Clv)

Remark 9.18 One can choose Alv, Blv, Clv to be explicit integers. See the proof below. For instance Clv ^ nG for nG defined in Sect. 1.8.

Proof Put $S:= 1Ks,~(n). The right hand side of (9.6) is expanded as in [3, Thm 6.1] as shown by Arthur. Arguing as at the start of the proof of [99, Thm 4.4], we obtain from Lemma 8.4 in view of the imposed lower bound on N (n) that

Hf,S1 (fS1) - Hf(fs) = Z aM • $S00,M(1)$S1,M(1)$M^(l)

M eLc (M0)\{G}

dim £

where the sum runs over proper cuspidal Levi subgroups of G containing a fixed minimal F-rational Levi subgroup (see [45, p. 539] for the reason why only cuspidal Levi subgroups contribute) and aM e C are explicit constants depending only on M and G. A further explanation of (9.9) needs to be given. Since only semisimple conjugacy classes contribute to Arthur's trace formula for each M, Lemma 8.4 tells us that any contribution from non-identity elements vanishes. Note that Hp(Is) comes from the M = G term on the right hand side.

The first assertion of the theorem follows immediately from (9.9). Henceforth we may assume that Lc(M0)\{G} = 0.

Clearly 0S0,M(1) and $M(1,%)/dim % are constants. It was shown in

Lemma 2.14 that I^,m(1)| = O(qS£+rG+bGK) for bG > 0 in that lemma. We take

Aiv := dG + rG and Blv := bG.

We will be done if it is checked that ^TO(1)I = O(N(n)-Clv) for some Clv ^ 1. Let P = MN be a parabolic subgroup with Levi decomposition where M is as above. Then

0 < ^MTO(1) = i , 4S,TO(n)dn JN (AFTO)

H vol(Kv(mV(n)) n N(Fv))

v|n or veRam(G)

J} vol( N (Fv)x ,v(n))

v|n or veRam(G) t \

—v(n) dim N

n vol(Kv n N(Fv)).

v e Ram(G)

v| v (

v /S v /S

The last equality uses the standard fact about the filtration that vol(N (Fv )x ,v(n) ) = Imv |v(n)dimN vol(N(Fv)x,0) and the fact (8.1) that vol(N(Fv)x,0) = vol(N(Fv) n Kv) = 1 when G is unramified at v. Take

Clv := min (dim N)

M e Lc (M0)\{G} P=MN

to be the minimum dimension of the unipotent radical of a proper parabolic subgroup of G with cuspidal Levi part. Then 4^TO(1)| < N(n)—'ClvUveRam(G) vol(Kv n N(Fv)) for every M in (9.9). □

9.5 Weight aspect

We put ourselves in the setting of Example 9.14 and exclude the uninteresting caseof G = {1}. By the assumption Z (G) = {1},forevery y = 1 e G (F) the connected centralizer IY has a strictly smaller set of roots so that | $ 1 | < | $ |. Our next task is to prove a similar error bound as in the last subsection.

Theorem 9.19 Fix 4s0 e CcTO(G(Fs0)) and US'TO c G(AS'TO). There exist constants Awt, Bwt > 0 and Cwt ^ 1 satisfying the following: for

any k e Z>0,

any finite subset S1 c VTO disjoint from S0 and Sbad (Sect. 7.2) and any 4s1 e Hur(G(Fs1 ))**" such that I4s1 I < 1 on G(Fs1),

fiFS fi) — fiS1 4) = o (qAwt+BwtKm(%)—Cwt)

where the implicit constant in O(•) is independent of k, S1 and 4S1. (Equiva-lently, (fiS1) — fipj(fi) = O(qSjwt+Bwtkm(%)—Cwt) if fig,) = 0.)

Remark 9.20 We always assume that S0 and S1 are disjoint. So the condition on S1 is really that it stays away from the finite set Sbad. This enters the proof where a uniform bound on orbital integrals from Sect. 7.2 is applied to the places in S1.

Remark 9.21 Again Awt, Bwt, Cwt can be chosen explicitly as can be seen from the proof below. For instance a choice can be made such that Cwt ^ nG for nG defined in Sect. 1.8.

Proof We can choose a sufficiently large finite set S0 D S0 U Ram(G) in the complement of S1 U STO such that US,TO is a finite disjoint union of groups of

the form ^v /S0US1USTO Kv) X US0\S0 for open compact subgroups US0\S0 of

G(AF,S^\S0). By replacing S0 with S0 (and thus S with S0 U S1), we reduce the proof to the case where US,TO = nv/SUSoo Kv.

For an F-rational Levi subgroup M of G, let YM be as in Proposition 8.7, where k , S0 and S1 are as in the theorem. (So the set YM varies as k and Si vary.) Take (9.6) as a starting point. Arthur's trace formula ([3, Thm 6.1]) and the argument in the proof of [99, Thm 4.11] show (note that our YM contains YM of [99] but could be strictly bigger):

fiF ,S1 (fiS1) — fiS1 (fis)

Z.G, M — 1r>G(ATO \ATOtr %n(Y) aG,Y • |t (Y)I OY f (4 )-7—T-

Ye Yg\{1} dim %n

+ E JlaMY ^|1M(Y)I—1OM(AF)(0TO)^^ (9.10)

MeLc\{G} Ye Ym lm %n

where aM,Y (including M = G) is given by

0-M ,y = T (G )

Cor 6.13 T Vy

A ^can'EP (¡M(F)\IM(Af)/AIyM (¡M(Fx)/AjM^ [¡M) \Qtm \ L(MotI M)

t'(G) ¡me^M1^ 2[F

Let us work with one cuspidal Levi subgroup M at a time. Observe that clearly ¡m\/\qim c\ ^ and that T(lf ) is bounded by a constant depending only on G in view of (6.3) and Corollary 8.12 or Lemma 8.13. By Corollary 6.17, there exist constants c2, A2 > 0 such that

\aM, Y \ ^ C2 Y[

veRam( IM)

It is convenient to define the following finite subset of V™ for each y e YM. We fix a maximal torus T f in M over F containing y and write , Y for the set of roots of Tf in M. (A different choice of Tf does not affect the argument.)

Sm,y := {v e V^\S: 3a e ,y, a(y) = 1 and \1 - a(y)\v = 1} .

(If y is in the center of M (F) then SM, Y = 0 and qSM = 1.)

We know that of {Fv)(1km,v) = 1 for v / S U Sm,y U Sx and that Sm, y D Ram(If) from [63, Cor 7.3]. According to Lemma 6.2 4>v = lKv implies $v,M = 1kMv . Hence

\aM,y \ < C2 ■ (qSM,Y)A2 (9.11)

of{kf\M = of™(4>S,M) n Of™(1Km,v).

veSu,y

By Theorem 13.1, there exists a constant c($So,M) > 0 such that

Of (FS0\$So,M) < c(fao,M) J] DM(y)-1/2, Vy e YM.

By Theorem 7.3, there exist a, b, c, eG e M^0 (independent of y, S1, k and k) such that

of(FS1 W,M) < qa:bKIl DM( Y)-eG/2, (9.12)

OM{Fv\lKMv) < qCvDM(Y)-eG/2, Vv e Sm,y (9.13)

[To obtain (9.12) and (9.13), apply Theorem 7.3 to v e S1 and v e Sm,y.] Hence

oM(AFM < c(0so,M)qas+bKqcSM,y ( n DM(Y)~1/2

x n DM (Y)(1-eG)/2 veSiUSM,Y

c(4so,M )qS+bK qSM^U DM (y)1/2

X n DM (y)(1-eG )/2 (9.14)

veSiUSM,Y

On the other hand there exist SSo, S^, SS1 ^ 1 such that for every y e YM with a(y) = 1,

- |1 - a(y)\s0 ^ Ss0. (compactness of supp4s0)

- |1 — a(y) ^ S^. (compactness of Ux)

- |1 — a(y)|Sl ^ SSl q^. (Lemma 2.18 and Remark 2.20 explains the independence of B1 of S1.)

[When a(y) = 1, our convention is that \1 — a(y)\v = 1 for every v to be consistent with the first formula of Appendix A.] Hence, together with the product formula for 1 — a(y),

1 = E[\1 — a(Y)\v < ^soSœSs1 qsB5^ n \1 — a(Y)\v

v veSMy

Set S := SSoSœSS1. Note that \1 — a(y)\v ^ 1 for all a e $My and all v e Sm ,y .If y e Z ( M )( F ) then qSMy = 1. Otherwise for each v e Sm,y ,wemay

choose a e $M,y such that \1 — a(y)\v = 1. Then \1 — a(y)\v ^ q—l/WGSG (for the same reason as in the proof of Proposition 8.7, Step I) so

qv < fa*")WGSG , v e sm, y (9.15)

In particular the crude bound maxveSMY qv ^ 2\ SM,Y \ holds, hence

\sm, Y \ < 2 ({sq*K)WGSG + 0 =: ^ (9.16)

Notice that the upper bound is independent of (and depends only on the fixed data). Keep assuming that y is not central in M and that a(y) = 1. Again by

the product formul^veS1USMY 11 - a( Y)\v = 11veSoUS11 - a( Y)|-1 ^

(Ss0$<x)-1, thus

H DM(y)-1 < <5S0<W (9.17)

veS1USM,Y

The above holds also when is central in M, in which case the left hand side equals 1.

Now (9.14), (9.15), (9.16), and (9.17) imply

Of(Af] №) < cnM)*CWGSGS'<W(eG-1)/2qa+bK+cB5WGSG>'"

x n DM(Y)1/2- (9.18)

Lemma 6.11 gives a bound on the stable discrete series character:

I$g (v t)| ff . dm( y)-1/2

(^ 1[vl™ Dv(y)+ . (9.19)

| i i i

dim £ |

m(£) ¡y

Multiplying (9.11), (9.18)and(9.19)altogether(and noting\iM ( y)\-1 < 1), the absolute value of the summand for y in (9.10) (including M = G) is

O (m dqa+b"+cB5wGSGs'k +a2^

All in all, ,s($s) - vf (4>s)\ is \ Yg\ - 1+ X \ Ym\] O (m(£)~( 1 *+1 -1 *+fl)qas+b"+cB5wG^'k+A2\.

MeLc\{G} / ^ '

Set (excluding y = 1 in the second minimum when M = G) Cwt := min min (|$+|-|$+ I)

M eL (Mo) y eM (F) 1y

ell .in M (Fx)

Note that Cwt depends only on G .It is automatic that | 1-| $ +M | ^ 1on

Yg\{1} and YM for M e Lc(M0)\{G}. The proof is concluded by invoking Corollary 8.10 (applied to YG and YM) with the choice

Awt := a + A2 + A6, Bwt := b + cB^wgSg+ B8.

9.6 Automorphic Plancherel density theorem

In the situation of either Examples 9.13 or 9.14, let us write Fk (4s0) for Fk in order to emphasize the dependence on 0So. Take S1 = 0 so that S = S0. Then ffFk (0S) , 0 may be viewed as a complex number (as it is a measure on a point). In fact we can consider Fk(fs), a family whose local condition at S is prescribed by fs e F(G(Fs)A), even if fs does not arise from any 4>s in CC°(G(Fs)). Put ffk(fs) := ff^k(f) 0 e C. We recover the automorphic Plancherel density theorem [99, Thms 4.3, 4.7].

Corollary 9.22 Consider families Fk in level or weight aspect as above. In level aspect assume that the highest weight of % is regular. (No assumption is necessary in the weight aspect.) For any fs e F(G(Fs)A),

lim ffk(Ts) = ff(Ts).

Proof Theorems 9.16 and 9.19 tell us that

lim ffk(fs) = f^s). (9.20)

(Even though there was a condition on S1, note that there was no condition on S0 in either theorem.)

We would like to improve (9.20) to allow more general test functions. What needs to be shown [cf. (9.21) below] is that for every e > 0,

lim sup

ffk ( fs ) - fpl( fs S)

Thanks to Proposition 9.1 there exist 0s, fs e Hur (G ( Fs )) such that | fs — fs I < fs on G( Fs )A and fiï§(fs ) < e. Then [cf. (9.22) below]

fk ( fs ) - ff( fs)

< \ fk ( fs - fs) l

fk (fS) - fS (fS) + fS (fS - fS)

Now \ff (fs - fs) \ < \ff (fs) \ < e, and \fk (fs) - ff (fs) \ < e for k » 1 by (9.20). Finally fk is a positive measure since the highest weight of £ is regular (see Example 9.11), and we get

\fk ( fS - fS ) ^ fk (\ fS - fS \ ) ^ fk (-%s )-

[To see the positivity of fk, notice that fk (fS - fS) is unraveled via (9.4) and (9.5) as a sum of (fS - fS)(n) with coefficients having nonnegative signs. This is because xEP (nTO® £) is either 0 or (- 1)q(G) when £ has regular highest weight, cf. Sect. 6.5.] According to (9.20), limk^TO fk(-fS) = ff (■fS) ^ e. In particular \ fk (fS - fS) \ ^ 2e for k » 1. The proof is complete. □

Remark 9.23 If G is anisotropic modulo center over F so that the trace formula for compact quotients is available, or if a further local assumption at finite places is imposed so as to avail the simple trace formula, the regularity condition on £ can be removed by an argument of De George and Wallach [38] and Clozel [22]. The main point is to show that the contribution of (£ -cohomological) non-tempered representations at to to the trace formula is negligible compared to the contribution of discrete series. Their argument requires some freedom of choice of test functions at to, so it breaks down in the general case since one has to deal with new terms in the trace formula which disappear when Euler-Poincare functions are used at to. In other words, it seems necessary to prove analytic estimates on more terms (if not all terms) in the trace formula than we did in order to get rid of the assumption on £. (This remark also applies to the same condition on £ in Sects. 9.7 and 9.8 for level aspect families.) We may return to this issue in future work.

Remark 9.24 In the case of level aspect families [99, Thm 4.3] assumes that the level subgroups form a chain of decreasing groups whose intersection is the trivial group. The above corollary deals with some new cases as it assumes only that N(nk) ^ to.

Corollary 9.25 Keep assuming that S1 = 0.Let(uf,TO, £k) = (KSTO(nk), £) or (US,TO, £k) in Examples 9.13 or 9.14, respectively, but prescribe local conditions at S by fs rather than ¿S. Then

fcan (ukS,TO)

lim -^—f-\Fk\ =fpJ( fS)■

k^TO T '(G) dim £k kl

Proof The corollary results from Corollary 9.22 since

-T^k| = -t Z^ afk (n) = ffk ,0 ( fs ).

T'(G) dim %k T'(G) dim %k tr,

n eardisc,xk (G)

9.7 Application to the Sato-Tate conjecture for families

As an application of Theorems 9.16 and 9.19, we are about to fulfill the promise of Sect. 5.4 by showing that the Satake parameters in the automorphic families {Fk} are equidistributed according to the Sato-Tate measure in a suitable sense (cf. Conjecture 5.9).

The notation and convention of Sect. 5 are retained here. Let 6 e C(T1) and Te F(%,e/&c,e). For each v e Vf(6), the image of .fin F(G(Fv)A'ur) via (9.1) will be denoted fv.

Theorem 9.26 (Level aspect) Pick any 6 e C(T1) and let {vj}jbe a sequence in VF (6) such that qvj as j Suppose that

- fps0 (fS0) = 0 and

- % has regular highest weight.

Then for every f e F (fc , 6 / , 6),

(j , ^ fFk , v j (f j) = fT ( f)

where the limit is taken over (j, k) subject to the following conditions:

T\TC \ — Bsmic ^ —1

- N(nk)qvj " > c—1,

- v j f nk,

- qNjN(nk)—1 ^ 0 for all N > 0.

Proof Fix f. We are done if lim sup(j,k)^<x> |fFk,vj (fj) — fST(f)| < 4e for

every e > 0. By Proposition 5.3, [f^ (fvj) — fST(f )| ^ e for sufficiently large j . So it is enough to show that

^ ' < 3e. (9.21)

lim sup

ffk vj ( f j) — fp ( fj)

For every j ^ 1, Proposition 9.4 allows us to find fa j ,fvj

fv j ) <

1 Springer

Hur (G (Fv j such that | fv j — fv j | < fv j on G (Fv j )A and (f j) < e.

For each j ^ 1,

fFk v j ( fvj ) - ftj ( fvj ) < fFk v j ( fvj — <t>vj )

V, (fv j ) - fpj (fv j )

>■k,vj^ v 1

(fv j — fv j )

(9.22)

Since ffl is a positive measure,

vl) (fv j — fv j )

^ (\fvj — fvj I) ^ fpj {fvj) ^

Theorem 9.16 and the assumptions of the theorem imply that for sufficiently large (j, k), \ ffp v. (fv j) - f(fv j) \ ^ e. So we will be done if for sufficiently large (j, k),

fFk,vj (fvj — fvj)

(9.23)

Arguing as in the proof of Corollary 9.22 we deduce the following: when fr, v(fvi — fvi) is unraveled as a sum over n [cf. (9.4) and (9.5)], each

Fk j J J

summandis fSo (nSo)(fvj — fvj )(nvj) times a nonnegative real number. (This uses the regularity assumption on £. Certainly the absolute value of the sum does not get smaller when every summand is replaced with (something greater than or equal to) its absolute value, i.e.

vj {fvj — fvj)

< fF af»v. (\fv j — fv j \) < (fs0)v j (f j )■

fk ( I Ws0 I ) , v j

Now choose 4>'So e CTO(G(FS0)) according to Lemma9.6 so that \$Sl0 (nS0) \ ^ ¿S0 (ns0) for every ns0 e G(Fs0)A. Then

fk ( I Ws01), v -

(fv j) ^ f

Fk W 0 j

(fv j )■

Theorem 9.16 applied to fv, and the inequality fvl (fv:) ^ e imply that

lim sup f ws ),vj

This concludes the proof of (9.23), thus also (9.21).

Theorem 9.27 (Weight aspect) Let 9 e C(r1) and ¿s0 e CTO(G(Fs0)). Suppose that {vj}jis a sequence in VF(9) such that qvj ^to as j ^ to

and that fff (fs0) = 0. Then for every f e F (Tc^9/&c,9),

(^Vj f j) = ^(D

if q N.m (Ik) 1 ^ 0 ask ^ to for any integer N ^ 1.

Proof Same as above, except that Theorem 9.19 is used instead of Theorem 9.16. □

Remark 9.28 As we have mentioned in Sect. 5.4, Theorems 9.26 and 9.27 indicate that {Fkare "general" families of automorphic representations in the sense of Conjecture 5.9.

Corollary 9.29 In the setting of Theorems 9.26 or 9.27, suppose in addition that |Fk | = 0 for all k ^ 1. Then

(.lim DFK f) = DST(?).

(j,k)^TO j

Proof Follows from Corollary 9.25 and the two preceding theorems (cf. Remark 9.9). □

Remark 9.30 The assumption that |Fk | = 0 is almost automatically satisfied. Corollary 9.25 and the a any sufficiently large k.

Corollary 9.25 and the assumption that '¡x1f0 (<Ps0) = 0 imply that |Fk | = 0 for

9.8 More general test functions at So

So far we worked primarily with families of Examples 9.13 and 9.14. We wish to extend Theorems 9.26 and 9.27 when the local condition at S0 is given by fS0, which may not be of the form </>Sq for any 0Sq e CTO(G(FS0)) (cf. Example 9.11 and Remark 9.12).

Corollary 9.31 Let 9 e C (T1) and let {vj}j be a sequence of places in VF(9) such that qVj ^to as j ^to. Consider fxFk,Vj where

f F (KS,TO(nk), fs0, v., level aspect, or Fk F (USTO, ffs0, v., Ik) weight aspect

satisfying the conditions of Theorems 9.26 or 9.27, respectively. Then

(jk£TO ^Vj (Dj) = ^(?)

where the limit is taken as in Theorem 9.26 (resp. Theorem 9.27).

Proof The basic strategy is to reduce to the case of 0 and 0Vj in place of f and fVj via Sauvageot's density theorem, as in the proof of Theorem 9.26. We can decompose f = f + + f_ with ff_ e F(fc,e/&c,e) such that f + and f_ are nonnegative everywhere. The corollary for f is proved as soon as it is proved for f + and f_. Thus we may assume that f ^ 0 from now on.

Fix any choice of e > 0. Proposition 9.1 ensures the existence of 0So, fs0 e

C£°(G (Fso)) such that fg, ( fo) < e and | fSo (nso) _ fSo (nso )l < fSo (nso)

forall nSo e G (FSo )A. Of course we can guarantee in addition that fif (fSo) = o. Put

Fk(fso) := F (KS,™(nk), fao, vj, & (resp. Fk(fSo) = F(Us,to, fso, Vj, &)) .

Likewise we define Fk (fgo) and so on. Then (cf. a similar step in the proof of Theorem 9.26)

Vj ( fVj ) - Vguivj} (fso fVj) < fFk (fso ),Vj ( fVj ) - U{v j }(fso fVj )

fFk (| fso —fSo l)( fVj ) + f poU{V j} (l-^o - fso \fVj)

The first term on the right side tends to o as (j, k) ^ to by Theorems 9.26 and 9.27. The last term is bounded by fgoU{ }( fSo fVj) ^ efj (fVj) using

the fact that ffg is a positive measure. In order to bound the second term, recall that we are either in the weight aspect, or in the level aspect with regular highest weight for &. Then aTk(f |)(n) is a nonnegative multiple of

l fSo (nSo) _ fSo (nSo )| as in the proof of Theorem 9.26. Thus

fFk (| f _fso l)( fvj) = l£Fk (| f _fso l)( fvj) ^ fFk (fso)( fvj) ^ (fSo)( fj

the last inequality coming from the bound ffg ( fgo) ^ e. Hence we have shown that

lim sup

fFk,Vj ( fVj ) _ fplou{vj}( fSo fVj )

^ e ^^ X (fso)(fj) + (fvj)

By Theorems 9.26 and 9.27 and the fact that lim f. ( fv.) = fSL ( f ), the

right hand side is seen to be bounded by 2effST ( f ). As we are free to choose e > 0, we deduce that

,. lim fFk ,v j( fj) = f0( f o )ffT ( A

Remark 9.32 It would be desirable to improve Theorems 9.16 and 9.19 similarly by prescribing conditions at S0 in terms of f0 rather than the less general fs0. Unfortunately the argument proving Corollary 9.31 does not carry over. For instance in the case of Theorem 9.16, one should know in addition that the multiplicative constant implicit in O (qAv+BlvKN(nk)-Clv) is bounded as a sequence of fS0 approaches fS0.

10 Langlands functoriality

Let r: LG ^ GLd(C) be a representation of LG. Let n e ARdisc,x(G) be such that with nv e ndisc(£V0 for each v\to (recall the notation from Sects. 6.4 and 9.2). The Langlands correspondence for G(Fv) [71] associates an L-parameter ^v : ^R ^ LG to the L-packet ndisc(^v/), cf. Sect. 6.4.

The following asserts the existence of the functorial lift of n under r as predicted by the Langlands functoriality principle.

Hypothesis 10.1 There exists an automorphic representation n of GLd (AF) such that

(i) n is isobaric,

(ii) nv = r*(nv) [defined in (2.9)] when G, r and n are unramified at v,

(iii) nv corresponds to rp^v via the Langlands correspondence for GLd(Fv) for all v\to.

If n as above exists then it is uniquely determined by (i) and (ii) thanks to the strong multiplicity one theorem. Moreover

Lemma 10.2 Hypothesis 10.1 (iii) implies that nv is tempered for all v\to.

Proof Recall the following general fact from [71, §3, (vi)]: let p be an L-parameter for a real reductive group and n(p) its corresponding L-packet. Then p has relatively compact image if and only if n (p) contains a tempered representation if and only if n(p) contains only tempered representations. In our case this implies that p^y has relatively compact image for every v \to, and the continuity of r shows that the image of r p^y is also relatively compact. The lemma follows. □

As before let (B, T, {Xa}aeAv) denote the Gal(F/F)-invariant splitting datum for G. Recall that Xty e X*(T)+ designates the highest weight for t'. Then pty I wC is described as

pty (z) = ((z/z)p+Xv, z) e G x Wc, Vz e Wc = Cx.

It is possible to extend pty I Wc to the whole of WR but this does not concern us. (The interested reader may consult pp. 183-184 of [65] for instance.) Let T be a maximal torus of GLd (C) containing the image r (T), and B a Borel subgroup containing T. Write r |g = ®ieIri as a sum of irreducible G-representations. For each i e I, denote by X(ri) e X*(T) the B-positive highest weight for ri. Write X(ri) = Xo(ri) + Y,aeA a(ri, a) • av for Xo(ri) e X*(Z(G))q and a(ri, a) e Q^o. Put IX(n)| := Y.aeA a(n, a) and

M($v) := max(a, Xtv), M(r) := max IX(ri)|.

aeA v ieI

Similarly define m (tv ) and m (r) by using minima in place of maxima. We are interested in the case where X0 (ri) is trivial for every i e I. This is automatically true if Z(G) is finite. (Recall that we consistently assume Z(G) = 1 in the weight aspect.)

Lemma 10.3 Suppose that X0 (ri) is trivial for every i e I. Hypothesis 10.1 (iii) implies that for each v

(2 + m(r)m(tv))11 < C(Uv) < (3 + 2M(r)M(tv))d.

In particular if Z(G) is finite, then the following holds for any fixed L-morphism r.

1 + m(tv) C(Uv) ^r M(tv)d

Proof First we recall a general fact about archimedean L-factors. Let p: WR ^ GLN(C) be a tempered L-parameter and decompose pIWc into GL1 -parameters as pIwc = ®k=1Xk. The archimedean L-factor associated with p may be written in the form [cf. (4.1)]

L(s, p) = rR(s - lk(p)). (10.1)

For each k assume that xk (z) = (z/z)ak for some ak e 1Z. Then we have for every 1 ^ k ^ N, ik(p) e 2Z^0 and, after reordering ik(p)'s if necessary,

IakI < Ilk(p)I < Iak| + 1. (10.2)

Indeed this comes from inspecting the definition of local L-factors as inof[102, 3.1, 3.3] for instance. (Use [102, 3.1] if ak = 0 and [102, 3.3] otherwise.)

Returning to the setup of the lemma, we have by definition L(s, Uv) = L(s, rFor each i e I we consider the composite complex L-parameter

V^v \wc ^ (ri ,1)

Wc -C G x Wc — GLdim ri (C)

decompose it as ©d/=a1ri xi, j. We can find ai, j e 2Z such that Xi, j (z) = (z/z)ai'j. For each i, the highest weight theory tells us that ait j = {p + A,£v , X(ri)) ^ 0forone j and layl ^ at, j forthe other j' = j .By(10.1)and (10.2), the analytic conductor for nv (introduced in Sect. 4.2) satisfies

d dim ri

C(Uv) = n(2 +|/Xk(nv)\) ^ n(3 + \ai,j\) k=1 ieI j=1

(3 + {p + X?V ,k(ri )))dim ri •

Further{p +X§v,X(n)) = {p,X(n)) + {X§v,X(n)) < \X(n)| + |X(ri)|M(&) < M(r)(1 + M(v). Hence

C(Uv) ^H((3 + M(r)(1 + M(£v)))dimri = ((3 + M(r)(1 + M(v))d•

Now we establish a lower bound for C(Uv). For each i, we apply (10.2) to

the unique j = j (i) such that ait j = {p + X^v, X(ri)). Then

C(Uv) (2 + \ai,j(i)l) = EI(2 + {P + X£V, X(ri)))

ieI ieI

> (2 + m(r)(1 + m(lv))f^

11 Statistics of low-lying zeros

As explained in the introduction an application of the quantitative Plancherel Theorems 9.16 and 9.19 is to the study the distribution of the low-lying zeros of families of L-functions A(s, U). The purpose of this section is to state the main results and make our working hypothesis precise.

11.1 The random matrix models

For the sake of completness we recall briefly the limiting 1-level density of normalized eigenvalues. We consider the three symmetry types G(N) = SO(2N), U(N), USp(2N). For each integer N ^ 1 these groups are endowed with their Haar probability measure. For all matrices A e G (N) we have a sequence &j = & j (A) of normalized angles [59]

0 < &1 < &2 < ••• < &N < N. (11.1)

Namely the eigenvalues of A e U(N) are given by e(&N) = e2i n&j1N. The eigenvalues of A e USp(2N) or A e SO(2N) occur in conjugate pairs and are given by e(±2&N).

The mean spacing of the sequence (11.1) is one. The 1-level density is defined by

Wsw(&) := ! V &(&j(A))dA.

JG(N) j ^N

The limiting density as N ^ to is given by the following [59, Theorem AD.2.2].

Proposition 11.1 Let G = U, SO(even) or USp. For all Schwartz functions & on R+,

lim Wg(n)(&) = f &(x)W(G)(x)dx,

where the density functions W (G) are given by (1.5).

The density functions W(G) are defined apriori on R+. They are extended to R_ by symmetry, namely W(G)(x) = W(G)(-x) for all x e R. For aPaley-Wiener function & whose Fourier transform $ has support inside (_1, 1), we have the identities

$(x )W (G )(x )dx =

$(0) if G = U,

$(0) + 2 $(0) if G = USp, (11.2)

$(0) - 2$(0) if G = SO(even).

11.2 The 1-level density of low-lying zeros

Consider a family F = (Fk )k^1 of automorphic representations of GL (d, AF). The 1-level density of the low-lying zeros is defined by

D(Fk; $) := ^ y y log C(Fk)) (11.3)

|Fk 1 niFk j v }

Here $ is a Paley-Wiener function; we don't necessarily assume $ to be even because the automorphic representations U e Fk might not be self-dual. See also the discussion at the end of Sect. 4.4. The properties of the analytic conductor C (Fk) ^ 2 will be described in Sect. 11.5.

Since $ decays rapidly at infinity, the zeros Yj (U) of A(s, U) that contribute to the sum are within O (1/ log C (Fk)) distance of the central point. Therefore the sum over j only captures a few zeros for each U. The average over the family U e Fk is essential to have a meaningful statistical quantity.

11.3 Properties of families of L-functions

Recall that in Sect. 9.3 we have defined two kinds of families F = (Fk)k^1 of automorphic representations on G(AF). The families from Example 9.13 are varying in the level aspect: N(nk) — to while the families from Example 9.14 are varying in the weight aspect: m (§k) — to. In both cases we assume that $s0 e CTO(G(Fs0)) is normalized such that

£S0 (&0) = = 1. (11.4)

For families in the weight aspect we assume from now the weights are bounded away from the walls. Namely we assume that we are given a fixed n > 0 and that

(dim §k)n < m (b), Vk. (11.5)

Given the continuous L-morphism r: LG — GL(d, C) we can construct a family F = r*F of automorphic L-functions. Assuming the Langlands functoriality in the form of Hypothesis 10.1, for each n e Fk there is a unique isobaric automorphic representation U = r*n of GL (d, AF). We denote by Fk = r*Fk the corresponding family of all such U. Recall from Sect. 9.2 that Fk is a weighted set and that the weight of each representation n is denoted aFk (n ). The same holds for Fk and in particular we have

lFkl = |Fk| = £ aTk(n).

We have seen in Corollary 9.25 that |Fk | — to as k — to.

By definition [see (9.4)], if n e Fk then nTO has the same infinitesimal character as , i.e. n e Udisc(§k). If U e Fk then UTO corresponds to the composition r o via the Langlands correspondence for GLd(FTO) [This is Hypothesis 10.1 (iii)]. In particular UTO is uniquely determined by §k and r. It is identical for all U e Fk.

It is shown in Lemma 10.2 that UTO is tempered. Therefore Proposition 4.1 applies and the bounds towards Ramanujan (4.6) are satisfied for all U e Fk.

To simplify notation throughout this and the next section, we use the convention of omitting the weight when writing a sum over Fk .If / (n) is a quantity that depends on n e Fk, we set

X /(n) : = X afk(n)/(r*n)-

neFk n efk

This convention applied in particular to (11.3) above.

11.4 Occurrence of poles

We make the following hypothesis concerning poles of L-functions in our families.

Hypothesis 11.2 There is Cpoie > 0 such that the following holds as k

# {n e Fk, Ms, n) has a pole} « |Fk\l-Cp°le.

The hypothesis is natural because it is related to the functoriality Hypothesis 10.1 in many ways. Of course it would be difficult to define the event that "L(s, n) has a pole" without assuming Hypothesis 10.1. Also when Functori-ality is known unconditionally it is usually possible to establish the Hypothesis 11.2 unconditionally as well. We shall return to this question in a subsequent article.

11.5 Analytic conductors

As in [53] we define an analytic conductor C (Fk) associated to the family. The significance of C (Fk) is that each n e Fk have an analytic conductor C(n) comparable to C(Fk). The hypothesis in this subsection will ensure that log |Fk \ x log C (Fk). We distinguish between families in the weight and level aspect.

11.5.1 Weight aspect

For families in the weight aspect we set C (Fk) to be the analytic conductor C(n m) of the archimedean factor n^ (recall that is the same for all n e Fk). Then C(n) x C(Fk) for all n e Fk.

From Corollary 9.25 we have that |Fk \ x dim as k ^ ro. It remains to relate the quantities C(Fk), dim£¡k and m(%k), which is achieved in (11.6) and (11.7) below.

Lemma 11.3 Let v|x. Let be an irreducible finite dimensional algebraic representation of G(Fv). Then m« dim« M(^)|Ф+1 Also M fa) « dim .

Proof This follows from Lemma 6.10. Recall the definition of m (%v) in Sect. 6.4 and M(%v) in Sect. 10. □

Because of (11.5) and the previous lemma we have that

m(&)|ф+| « dim & « m(&)1/n. (11.6)

From Lemma 10.3 we deduce that there are positive constants C1, C2 such that

m ($k)C1 « C(Fk) « m (&)C2. (11.7)

11.5.2 Level aspect

For families in the level aspect the situation is more complicated mainly because of the lack of knowledge of the local Langlands correspondence on general groups and the depth preservation under functoriality. We define C (Fk) by the following

log C(Fk) X log C(П),

|Fk| Пей

and we introduce the following hypothesis. Hypothesis 11.4 There are constants C3, C4 > 0 such that

N(nk)C3 « C(Fk) « N(nk)C4.

11.6 Main result

We may now state our main results on low-lying zeros of the family F = rxF. The following is a precise version of Theorem 1.5 from the introduction [compare with (11.2)].

Theorem 11.5 Assume Hypothesis 10.1 for individual representations as well as 11.2 and 11.4. There is 0 < 8 < 1 such that for all Paley-Wiener functions Ф whose Fourier transform Ф has support in (-8,8) the following holds:

^ s (r ) lim D(Fk, Ф) = Ф(0) - 4rФ(0),

k—x 2

where s (r) e {—1, 0, 1} is the Frobenius-Schur indicator of r: LG — GLd (C).

12 Proof of Theorem 11.5

The method of proof of the asymptotic distribution of the 1-level density of low-lying zeros of families of L-functions has appeared at many places in the literature and is by now relatively standard. However we must justify the details carefully as families of L-functions haven't been studied in such a general setting before. The advantage of working in that degree of generality is that we can isolate the essential mechanisms and arithmetic ingredients involved.

In order to keep the analysis concise we have introduced some technical improvements which can be helpful in other contexts: we use non-trivial bounds towards Ramanujan in a systematic way to handle ramified places; we clarify that it is not necessary to assume that the representation be self-dual or any other symmetry property to carry out the analysis; most importantly we exploit the properties of the Plancherel measure when estimating Satake parameters. Previous articles on the subject rely in a way or another on explicit Hecke relations which made the proof indirect and lengthy, although manageable for groups of low rank.

12.1 Notation

To formulate the main statements we introduce the following notation

— 1 i'TO L' /1 \

CkA y) : = ^ Z / — ( - + ix,nA e2n iyxdx, v e Vf , y

¡Fki Jl v2 )

(12.1)

We view Lk,v as a tempered distribution on R. Note that when v is non-archimedean £k,v is a signed measure supported on a discrete set inside R>0.

The proof of the main theorems will follow by a fine estimation of Ckv( y) as k ^ to. The uniformity in both the places v e VF and the parameter y e R will play an important role. Typically qv will be as large as C(Fk)O® and y will be of size proportional to log C (Fk).

The first step of the proof consists in applying the explicit formula (Proposition 4.4). There are terms coming from the poles of L(s, n) which we handle in Sect. 12.4. The second term in the right hand-side in Proposition 4.4 is expressed in terms of the arithmetic conductor q (n) and will yield a positive contribution in the limit for families in the level aspect. When evaluating the 1-level density D(Fk, &) it remains to consider the following sum over all places

k>z y '>■$( logCD)' <12-2)

log C (Ft ' Mog C (Ft)

plus a conjugate expression, see Sect. 12.3.

Our convention on Fourier transforms is standard. Let $ be a Schwartz function on R. The Fourier transform is as in <4.7) and the inverse Fourier transform reads

$( y )e2n ixydy.

Given two Schwartz functions $ and ^ we let

$(x )V(x )dx.

Sometimes we use the notation <$ (x), ^ (x)) to put emphasize on the variable of integration. The Plancherel formula reads

<$(x ),*(x )) = <$( y ),$(-y)). <12.3)

We use the same conventions for tempered distributions. The Fourier transform of the pure phase function x — e2'nax is the Dirac distribution S(a) centered at the point a. To condense notation we write

*(y) := $

( 2n y \ \log C (Ft))

and shall express our remainder terms with the quantities || ^ || x ^ || $ || and || $ || 1 ^ II $ || 1. Since $ is fixed these are uniformly bounded, independent of

k —> x.

There are different kinds of estimates depending on the nature of the place v e VF. We shall distinguish the following set of places:

(i) the archimedean places Sx, the contribution of which is evaluated in Sect. 12.5;

(ii) a fixed set So of non-archimedean places. These may be thought of as the "ramified places". Their contribution is negligible as shown in Sect. 12.7;

(iii) the set {v | nk} of places that divide the level. These play a role only when the level varies and we show in Sect. 12.10 that their contribution is negligible. We use the convention that for families in the weight aspect this set of places is empty;

(iv) the generic places Sgen which is the complement in VF of the above three sets of places. This set will actually be decomposed in two parts:

Sgen — Smain U Scuti

(v) where the set Scut is infinite and consists of those non-archimedean places v e Sgen such that loq is large enough to be outside of the support of ^ [see (12.18) below for the exact definition of Scut]. Then the pairing in (12.2) vanishes;

(vi) the remaining set Smain is finite (but growing as k — to). It will produce the main contribution of (12.2). For all places v e Smain, each of G, r and n is unramified over Fv. Using the notation of Sect. 5 we split Smain further as the disjoint union of

Smain n Vf(0), 0 e C(V1).

12.2 Outline

For non-archimedean places v e Smain we study in Sect. 12.6 various moments of Satake parameters. The quantity Lpl,v in (12.11) below will be the analogue of (12.1) where the average over automorphic representations n e Fk gets replaced by an average of nv against the unramified Plancherel measure. Our Plancherel equidistribution theorems for families (Theorems 9.16 and 9.19) imply that £k,v is asymptotic to Lpl,v as k — to.

It is essential that our equidistribution theorems are quantitative in a strong polynomial sense. Details on handling the remainder terms are given in Sects. 12.8-12.10.

For the main term we then need need to show the existence of the limit of

-1- T (Cplv,V) (12.4)

log C (Fk) ^ 1 P ' 1

ve Smain

as k —> to. The evaluation of Lpl,v is a nice argument in representation theory, see Sect. 12.12 where we shall see clearly the role of the two assumptions on r (that r is irreducible and does not factor through WF). The evaluation of Lpl,v can actually be quite complicated since it depends on the restriction of r to subgroups G x WFv for varying v e Smain and on the Plancherel measure on G(Fv)A'ur. Fortunately the expression will simplify when summing over all places v e Smain and applying the Cebotarev density theorem (see Sect. 12.11).

The overall conclusion of the below analysis is that the limit of (12.4) as k — to is equal10 to _ ^ $ (0), where s(r) is the Frobenius-Schur indicator

10 A quick explanation for the minus sign is as follows. A local L-factor is of the form (1 _ aq_s)-1 with three minus signs thus its logarithmic derivative is _ logq ^v>1 avq_vs with one minus sign.

of r .In the derivation of the one-level density there is an additional term $ (0) which easily comes from the explicit formula and the contribution of the archimedean terms. Thereby we finish the proof of Theorem 11.5.

12.3 Explicit formula

We apply the explicit formula (Proposition 4.4) for each n e Fk to obtain

$(0) z log q (n) z —

D(Fk, *) = Dpol(Fk, y , ^VJT Dv(Fk, $)+Dv(Fk,

|Fk1 neFklog c(Fk) viVP

(12.5)

Here Dpoi(Fk, $) denotes the contribution of the eventual poles. Also we have set

*> := Z H L G+nv) H log c o

2n |Fk 1 nli/-» L

See also the remark in (4.9) explaining how to shift contours. The scaling factor log2Cfk) comes from (11.3).

Applying Fourier duality (12.3) and the definition (12.1) implies the equality11

Dv(Fk, = * (LkAy), Mv^T^))■ (12.6)

log C (Fk )\ Vlog C (Fk)//

We have made a change of variable so as to make explicit the multiplicative factor 1 / log C (Fk) in front of the overall sum. Similarly we have

kr z JZ L (2+2niog c (Fk 0

1Fk)( log C(Fk

log C (Fk ) 12.4 Contribution of the poles

The contribution of the poles in the explicit formula above is given by

Dpol(Sk.♦) :=iF-y ZZ *(^ log C<F))■

m k 1 neFk j y J

11 Note that the exponential in (12.1) is e2inxy with a plus sign.

We bound the sum trivially and obtain

, „ ^ * in e F. L(F n)hasapole) c F )O v

where the last term comes from the exponential order of growth of $ along the real axis because the Fourier transform $ is supported in (-S, S).

12.5 Archimedean places

In this subsection we handle the archimedean places v e Sx. Recall from Lemma 10.2 that n^ is tempered. In fact we shall only need here a bound towards Ramanujan 0 < 6 < 1 as in Sect. 4.2.

Lemma 12.1 For all f e C with ^e f ^ 6, and all Schwartz function ^, the following holds uniformly

fœ Y' (1 \ ^ /1 \

I Y (2-^+ix) *(x)dx=*(0)i°M 2 - n+0yi+ yx)yi}

Proof We have the following Stirling approximation for the Digamma function [traditionally denoted f(z)V-

Y(z) = log z + O (1) (12.7)

uniformly in the angular region |arg z I ^ n — e, see e.g. [51, Appendix B]. Since 6 < 1/2 all points 1 — x + ix lie in the interior of the angular region and we can apply (12.7). We note also that uniformly

log Q — X + ix^ = log Q — ^ + O(log(2 + Ix|)),

and this conclude the proof of the proposition. □

Remark 12.2 Note that the complete asymptotic expansion actually involves the Bernoulli numbers and is of the form

Y 1 ^ B2n / 1 \

Y(z) =log z + 2Z ^ 2^ + 0(z2Ti«) • (128)

From (12.8) we have that

y(a + it) + y(a — it) = 2 y(a) + O ((t/a)2)

holds uniformly for a and t real with a > 0. As in [53, §4] this may be used when the test function ^ is even (e.g. which is the typical case when all representations n e F are self-dual). We don't make this assumption and therefore use (12.7) instead.

Corollary 12.3 Uniformly for all archimedean places v e Sl and all Schwartz function ^, the following holds

Lk,v, =x x logv (2 - i nv))+o (n$ni).

IFk 1 neFki=1 ^ '

Here we have set logv z : = ^ log z when v is real and logv z : = log z when v is complex.

Proof Recall the convention (4.1) on local L-factors at archimedean places v e Sx. From Fourier duality (12.3) and the definition (12.1) we have

xj-l L G+ix'nv)$(x )dx-

Note that

f(s ) =

— 1 log n + 1(2), when v is real,

— log(2n) + ), when v is complex.

Applying Lemma 12.1, the estimate in the corollary follows. Recall from Proposition 4.1 that the bounds towards Ramanujan in Sect. 4.2 apply to all

n e Fk. □

We may continue the analysis of the contribution of the archimedean places to the one-level density. For v e Sœ, the local L-function L(s, nv) are the same for all n e Fk. We therefore conclude that

Z Dv(Fk,$) + Dv(Fk

vesœ ^ — — log C(Fk)

i,veSœ i=1 $(0)

2 — Mi (nv)

+ O (1)

Z Z2logv

Zlog C (nv) + O (1)1 . (12.9)

logC(^k) \veSx

In the last line we used the definition of the analytic conductor at archimedean places from Sect. 4.2.

12.6 Moments of Satake parameters

Now let v e V^ be a non-archimedean place. A straightforward computation shows that

-(s, Uv) = logq^ß(v)(Uv)q-

where ^(v)(n„) := a1 (nv)v +----+ ad(nv)v. Averaging over the family F

we let

^(Fk ) := t^X P(v)(nv)> v e > 1.

IFk 1 neFk The formula (12.1) becomes

Ck,v = - log q^ /3(v) (Fk)q-v/2s(V log , (12.10)

where S is Dirac distribution (see Sect. 12.1). Similarly for all v e 5"gen we let

£pl,v := - log q^^PVi q-V/2^ V lQg qv)> (12.11)

where the coefficients are defined locally as follows. Since v e 5gen, the

I(}^WFv

L-morphism, i.e. it factors through G x W™. Recall from Sect. 3 that gpl,ur

group G is unramified over Fv and that the restriction r | g x Wp is an unramified L-morphism, i.e. it factors through G x WFv. Recall from Sect. 3 1 is the restriction of the Plancherel measure g to G(Fv)A,ur. Then

:= r*Y + ■■■ + YV)), (12.12)

where we are using the convention in Sect. 2.3 for the L-morphism of unramified Hecke algebras r* : Hur(GLd(Fv)) — Hur(G(Fv)) and the Satake isomorphism with the polynomial algebra in Y1,...,Yd (Sect. 2.4).

The supports of both measures and £pl,v are contained in the discrete set l°gq£N^1. If qv is large enough this is disjoint from the support of ^ and thus all sums over places v e VF considered below shall be finitely supported.

12.7 General upper-bounds

Recall from Proposition 4.1 that the bounds towards Ramanujan apply to every n e Fk. Thus for every non-archimedean v e VF°, we have the upper bound \ai(nv)| ^ ql from which it follows that for every v ^ 1,

Pvv)(Fkdqf. Proposition 12.4 (i) For all v e VF° and all continuous function *,

($k,v,y) « qv 2 log qv II* .

(ii) For all v e Sgen and all continuous function *,

^ _ 1 ($pl,v,*) « qv 1 log qv II* II œ .

Proof (i) Inserting the above upper bound into (12.10) we have

Lkv, *) « log qv X qVv(e—1/2) (2n log qv) \ .

Because 0 < 6 < 2, the conclusion easily follows. (ii) The Plancherel measure /xpl,ur has total mass one and is supported on the tempered spectrum G(Fv)A'ur'temp (see Sect. 3.2). We deduce similarly that for every v ^ 1,

PpvUv < d (12.13)

Indeed the image of any unramified L-parameter r o p : WF ^ GLd (C) is bounded and all Frobenius eigenvalues have therefore absolute value one. □

12.8 Plancherel equidistribution

We are in position to apply the Plancherel equidistribution theorem for families established in Sect. 9. We shall derive uniform asymptotics as k ^ i for

pvv)(Fk).

Proposition 12.5 There exist constants C5 > 0 and A7, B9 < i such that the following holds uniformly on v ^ 1 and v e Sgen

pvv)(Fk) = (1 + oW)^ + O (qA7+B9vC(Fk)-C5) . (12.14)

Proof Let So be a sufficiently large set of non-archimedean places which contains all places v e V? where G is ramified and where r is ramified. Let S1 := {v}. We set

:= r*(YV + ■■■ + Yvd) e Hur(G(Fv)).

The notation for the Satake isomorphism is as in Sects. 2.2 and 2.5. By definition we have that = (^X Thanks to Lemma 2.6 we have that e Hur(G(Fv))^Pv and \$v| < 1. Thus we are in position to apply the respective Theorems 9.16 (in the level aspect) and 9.19 (in the weight aspect). Using the notation of Sect. 9.2, we have by construction

pvv)(Fk) = X aTk(n)*v(*v)

--county ) r'(G) dim & ^ .

k /¿can(UkS'?) \Fk\ k

r '(G) dim tk

The Corollary 9.25 shows that-f-= 1 + o(1) as k — We

/xcan(UkS'?) \Fk\

shall now distinguish between the two types of families.

For families in the level aspect, the assumption (ii) in Theorem 9.16 is satisfied because v \ nk for all k and all v e Sgen. If the assumption (i) in Theorem 9.16 is not satisfied, then

_ . Bt mK

C (Fk)C4 < N (nk) < csqv

where the first inequality comes from Hypothesis 11.4. Thus the error term in (12.14) dominates if A7 is chosen large enough. If the assumption (i) in Theorem 9.16 is satisfied, then from (9.8) we obtain the main term in (12.14) and the error term O(q^+BlvKN(nk)-Clv). By Hypothesis 11.4 we may then choose then C5 := Civ/C4 to conclude the proof of (12.14).

For families in the weight aspect the assumptions in Theorem 9.19 are always satisfied. This yields the main term in (12.14) with the error term O(qAwt+BwtKm(tk)-Cwt). By the estimate (11.7) we may choose C5 := Cwt / C2 to conclude the proof of (12.14). □

12.9 Main term

We deduce from Proposition 12.5 the following estimate for Ckv

Proposition 12.6 For all A > 0 there is A8 > 0 such that the following holds uniformly for all v e Sgen and all continuous function V:

[LKv, V) = (Lpi,v, V)(1+o(1)) + O (qA8 C(Fk)-C5 IIV ID + O(q—A IIV II to),

Proof Let k be a large enough integer. We apply the bounds towards Ramanu-jan in the form (12.7) to those term in (12.10) with v > k. The contribution of those terms to [Ck,v, V) is uniformly bounded by

« qf~1) IIV Wto •

We have that A := k(2 — 6) may be chosen as large as we want since 6 < 2 is fixed and k is arbitrary large.

For those terms in (12.10) with v ^ k we apply (12.14). Their contribution to (£k,v, V) is equal to

—logqv 2 P(£vq—v/2v (v lQg qv) + O {qA7+B9K iivIL cf)—C5) •

The next step is now to complete the v-sum. Applying (12.7) we see that the terms v > k yield another remainder term of the form q—A || V || to with A arbitrary large (again depending on k). □

12.10 Handling remainder terms

In this subsection we handle the various remainder terms and show that they don't contribute to D(Fk, 4>) in the limit when k ^ to. We shall apply the above estimates to the function

V(y):=¡ogCy-y e R (1215)

Recall from (12.6) that Dv(Fk, &) = (Lk,v, V)/log C(Fk). For archimedean places v e Sto we encountered in Sect. 12.5 the remainder term O(IVII1). Because log C(Fk) ^ to, this remainder term is negligible for Dv(Fk, 4>) as k ^ to.

For the non-archimedean places v such that v | nk or v e S0 we use the general bounds of Sect. 12.7 that imply

^ ^ ^ 6- 1 X Lk,v,V) « X qv 2 log qv WV ||to « 1 + # {v I nk} •

veS0, and v|nk veS0, and v|nk

(12.16)

In the last inequality we used the fact that S0 is fixed and that 6 < 2. Again the multiplication by 1/ log C (Fk) shows that these terms are negligible for D(Fk, 4>) as k — Indeed it is easy to verify that

# {v \ nk} = o(log N(nk)), as k — (12.17)

and we conclude using Hypothesis 11.4 that this is o(log C(Fk)).

We partition the set of generic non-archimedean places Sgen into two disjoint sets Smain and Scut where

Scut := {v e Sgen: qv > C(Fk)} • (12.18)

Since the support of $ is included in (S, S) we know that ^(v log qv/2n) vanishes for all v e Scut and v ^ 1.

For the generic places v e Smain we use the estimate in Proposition 12.6. The second remainder term yields

Z1 q-A ||„ = O\-1-(12.19)

logCKVKJ CS Hv 11 11 ? VlogC(Fk)/

veSmain

This is again negligible as k — The first remainder term in Proposition 12.6 is negligible as well because

Z qA8II* IIx c(Fk)-C5 « C(Fk)s(A8+1)-C5 (12.20)

_ IV v e Smain

and S is chosen small small enough such that S(A8 + 1) < C5.

Finally we show that the contribution to ( £pl,v, * ) / log C (Fk ) of the higher moments v ^ 3 is negligible. Because of the definition (12.11)of £pl,v and the bound (12.13) for ^, the contribution of the higher moments is uniformly bounded by

Z log qv Z q-v/2^^) « II* II» z q-3/2log qv « 1.

ve^main v^3 \ n / veV»

(12.21)

Therefore we can write the main contribution to D(Fk, $) as

-- Z (gpiv,*) = M(1) + M(2) + o(-1-

log C(Fk) ^ 1 Pl / Vlog C(Fk)J

vesmain

where for v = 1, 2 we define

Mv = - y q-v/2) . (12.22)

^ log C(Fkyv Vlog C(Fk)J

ve Smain

[recall the relation (12.15) between $ and ^]

12.11 Sum over primes

It remains to estimate the above terms (12.22) which consist of sums over the places v e Smain. We shall use the prime number theorem and the Ceb-otarev equidistribution theorem which we now proceed to recall, following e.g. [79, Chap. 7]. Let E/F be a finite Galois extension with Galois group f = Gal(E/F). For all conjugacy class 6 e C(O, recall that VF(6) consists of those unramified places v e V? such that Frv e 6.

Proposition 12.7 (Prime number theorem) Notation being as above,

# (v e VF?, qv ^ x) ---, as x

F log x

(Cebotarev equidistribution theorem) For any 6 e C (f),

# {v e VF(6), qv ^ x}---x —, as x

logx \f\

As a corollary we deduce the following estimate for any 6 e C (f)

^ logqv -1 logqv \ (\6\ . n,\

x / $y)dy, as k (12.23)

This estimate will be used below to evaluate M(2).

Note that if we replace log qv by — log qv, the same estimate holds with the integral on the right-hand side ranging from to 0. We shall use this observation below when adding the contribution of Dv (Fk, $) which will then produce produce the integral $(y)dy = $(0).

12.12 Computing the moments M(1) and M(2)

Recall that by assumption

r : G x Gal(F/F) = LG — GLd(C)

is irreducible and does not factor through Gal(F/F).

Lemma 12.8 The restriction r | g does not contain the trivial representation.

Proof If there were a non-zero vector in Cd invariant by r (G) then all its translates by Gal( F / F ) would still be invariant because G is anormal subgroup of LG. Because r is irreducible these translates generate Cd and thus the restriction r | g would be trivial12 which yields a contradiction. For an extension of this argument see e.g. [93, Prop. 24, § I.8.1]. □

Since v e Smain, the group G is unramified over Fv and the restriction r |gx Wp

is an unramified L-morphism which factors through G x W^. Note that this restriction might reducible in general.

Let A is a maximal Fv-split torus and QFv the Fv-rational Weyl group for (G(Fv), A). Recall from Sect. 2 the Satake isomorphism

5 : Hur(G) — Hur(A)iiFv.

For the group GLd the right hand-side is identified with C[F±,... Y±]Sd. We recall the morphism of unramified Hecke algebras r* : Hur (GLd) — Hur(G(Fv)) and the test functions:

$vv) := r*(Yv + ••• + Yvd) e Hur(G(Fv)).

In view of (12.12) we have

=gpl {gvv))=*vv)(1)-

Proposition 12.9 The following estimate holds uniformly for all v e S,

PÏÏv = O (q-1) -

Proof We decompose the restriction of r to G x into a direct sum of irreducible ©¡r;. By Lemma 12.8 each ri \g does not contain the trivial representation. In particular each n does not factor through .

12 In the sense that r\g would be a direct sum of trivial representations. In the sequel we use this slight abuse of notation when saying that a representation is "trivial".

We can now apply Lemma 2.9 which shows that

^ qvl Ifi fv \ max p(Xi *w 0). wefipv

Here Xi is as defined in Sect. 2.2. The two terms |^Fv| and p(X; *w 0) are easily seen to be bounded (uniformly with respect to v e Smain). D

As a consequence of Proposition 12.9 we deduce that

M(1) = O (-1-| (12.24)

Vlog C (Fk )J

—3/2

because the summand over v in (12.22) is dominated by qv .

For the second moment M(2) we shall need a more refined estimate. Recall the finite extension F1/ F from Sect. 5. We also choose a finite extension F2/ F\ such that r factors through G x Gal(F2/F). Let T2 := Gal(F2/F) and denote by C(r2) the set of conjugacy classes in r2.

Proposition 12.10 (i) For all 9 e C(T2) there is an algebraic integer s (r,9) such that uniformly for all v e Smain,

= s(r, [Frv]) + o (q-1) . (12.25)

Here [Frv] e C(T2) is the conjugacy class of Frv in T2. (ii) The following identity holds

s (r) = Z TpT s (r,9)

9 eC (r2) 1 21

where s(r) e {-1, 0, 1} is the Frobenius-Schur indicator ofr.

Proof (i) We proceed in way similar to the proof of Proposition 12.9 above. We shall give an explicit formula (12.26) for s (r, 9).

We decompose Sym2 r = ®p+ (resp. /\2 r = ®p-) into a direct sum of irreducible representation of G x Gal(E/F). Then we can decompose for each

i the restriction p+\q*a Wur =®j P+ as a direct sum of irreducible representa-

tions of G x WF. Similarly we let P-\qaWm = ®jP-.

Let := (p+)*(Y2 + ■ ■■ + Yj,) and similarly for Then it is easily verified that

$v2) = Z -T 9-j.

We now distinguish two cases. In the first case, i is such that p+ does not factor through Gal(E/F). Then by Lemma 12.8 the restriction p+\g does not contain the trivial representation. Thus for all j, p+\g does not contain the trivial representation. In particular p+ does not factor through W^. By Lemma 2.9 we deduce that $+(1) = O (q—1). These representations p+ only contribute to the error term in (12.25).

In the second case, i is such that p+ does factor through Gal(E/F). Then for all j, p+ factors through WF (in particular it is 1-dimensional). We have that g+(1) = p+(Frv). By linearity we deduce that Xj $+(1) = tr p+(Frv). This is an algebraic integer which depends only on the conjugacy class of Frv in

We proceed in the same way for $-. We deduce that (12.25) holds with 0 = [Frv] and 5(r, 0) = 5+(r, 0) — 5~(r, 0), where

^+(r,6) := Z tr p+(6) (12.26)

p+ factors through Gal( E / F)

and similarly for the definition of 5 (r,6). This concludes the proof of assertion (i).

(ii) By orthogonality of characters we have for each i such that p+ factors through Gal( E / F),

Z TF1 tr P+(6) = <l,P+> =

6 eC (r2) We deduce that

V \r2j

1, if p+ = 1 0, otherwise.

Z 5+(r,6) = <1, Sym2r>,

6 eC (r2)

the multiplicity of the trivial representation 1 in Sym2 r [as a representation of G x Gal(E/F)]. The same identity holds for 5—(r, 0) and /\2 r. From the definition of the Frobenius-Schur indicator 5 (r) in Sect. 6.8 we conclude the proof of the proposition. □

As a corollary we have the following estimate for the second moment:

M(2) = — y 5 (r,0) y log qv q—1g

¿—t v ' 2 ¿—t log C(Ft

eeC(F2) veSmainnVF (6) 5 w k

\log C (Fk )) \log C (Fk ))

We can extend the sum to v e Sgen n VF (9) because

log qv -, loglogC(F)

log C (Fk yv log C (Fk )

uniformly as k ^ œ. Applying the Cebotarev equidistribution theorem we deduce that

M(2) = - ^ 5 (r, 9^ y^l + °(1)) IT y)dy

9 eC (f2)

= {- sJT + *»)

The last line follows from Proposition 12.10 (ii) above.

2 + °(l^o y )dy. (12.27)

12.13 Conclusion

We now gather all the estimates and conclude the proof of Theorem 11.5. The explicit formula (12.5) expresses D(Fk, 4>) as the sum of four terms. The term Dpoi(Fk, $) goes to zero as k as consequence of Hypothesis 11.2, see Sect. 12.4.

The archimedean terms are evaluated in (12.9). In addition with the second term in (12.5) which involves log q (n), these contribute

$(0) ^ log C(n)

- 7--h °(1)-

IFk I log C (Fk ) neFk

This is equal to $(0) + o(1) (using the Hypothesis 11.4 for families in the level aspect).

We now turn to the non-archimedean contribution. The places v e S0 and v I nk are negligible thanks to (12.16) and (12.17), respectively.

It remains the non-archimedean places v e Sgen = Smain u Scut. The contribution from v e Scut is zero because the support of $ is included in (-S, S), see (12.18).

For each v e Smain we apply Proposition 12.6. The sum over v e Smain of the remainder terms is shown to be negligible in (12.19) and (12.20). For the main term the estimate (12.21) shows that the contribution of the higher moments is negligible. It remains the two terms M(1) and M(2) as defined in (12.22).

The asymptotic of M(1) and M(2) are given in (12.24) and (12.27) respectively. There is a similar contribution from the conjugate Dv(Fk, 4>). Overall this yields

veVp — Smain

^ — s (r) f™^

X Dv(Fk, $) + Dv(Fk, $) = J y)dy

v e ^main

s(r) f° ^ s(r)

<£( y )dy + o(1) = —^ $(0) + o(\).

We can now conclude that

^ 5 (r)

lim D(Fk, $) = $(0) - $(0). This is the statement of Theorem 11.5. □

Acknowledgments We would like to thank Jim Arthur, Joseph Bernstein, Laurent Clozel, Julia Gordon, Nicholas Katz, Emmanuel Kowalski, Erez Lapid, Jasmin Matz, Philippe Michel, Peter Sarnak, Kannan Soundararajan and Akshay Venkatesh for helpful discussions and comments. We would like to express our gratitude to Robert Kottwitz and Bao Chau Ngo for helpful discussions regarding Sect. 7, especially about the possibility of a geometric approach. We appreciate Brian Conrad for explaining us about the integral models for reductive groups. We thank the referee for a very careful reading. Most of this work took place during the AY2010-2011 at the Institute for Advanced Study and some of the results have been presented there in March. We thank the audience for their helpful comments and the IAS for providing excellent working conditions. S. W. S. acknowledges support from the National Science Foundation during his stay at the IAS under Agreement No. DMS-0635607 and thanks Massachusetts Institute of Technology and Korea Institute for Advanced Study for providing a very amiable work environment. N. T. is supported by a Grant #209849 from the Simons Foundation. We gratefully acknowledge BIRS, and the organizers of the workshop on L-packets, where this appendix was conceived. J. G. is deeply grateful to Sug-Woo Shin, Nicolas Templier, Loren Spice, Tasho Statev-Kaletha, William Casselman, and Gopal Prasad for helpful conversations. R.C. was supported by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement No. 615722 MOTMELSUM and by the Labex CEMPI (ANR-11-LABX-0007-01); J. G. was supported by NSERC; I. H. was supported by the SFB 878 of the Deutsche Forschungsgemeinschaft.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Appendix A: By Robert Kottwitz

Let F be a finite extension of Qp, and G a connected reductive group over F. For each semisimple y e G (F), define a positive real number

DG (y)

det(l — Ad(Y) iLie G/Lie G0 )

H |1 — a(Y)iv. (13.1)

ae$ a(y)=1

[In particular if y belongs to the center of G(F) then DG(y ) = 1.] We equip G (F), as well as IY(F) (the connected centralizer of y) for each semisimple

Y e G (F), with the Haar measures as in [47, §4]. The quotient measure on IY (F)\G(F) is used to define the orbital integral OY (f).

Theorem 13.1 For each f e C^°(G(F)), there exists a constant c( f) > 0 such that for all semisimple y e G (F),

|Oy(f )l < c(f) • Dg(y)-1'2.

Proof There are only finitely many G(F)-conjugacy classes of maximal F-tori in G, so in proving the theorem we can fix a maximal F-torus T in G and restrict attention to elements y lying in T (F). Then we must show that the function y — DG(y)1'2Oy( f) is bounded on T(F). Harish-Chandraproved that the restriction of this function to the set of regular elements in T (F) is bounded, so we just need to check that his methods can be used to treat singular

Y as well.

Since the function y — DG (y)1'2 Oy( f) is compactly supported on T (F), it is enough to show that it is also locally bounded. Harish-Chandra's method of semisimple descent reduces us to proving local boundedness in a neighborhood of 1 e T (F), and then the exponential map reduces us to the analogous problem on the Lie algebra g of G. The remainder of this appendix handles g, the main result being Theorem 13.11. □

A.1 Notation pertaining to the Lie algebra version of the problem

We write t for the Lie algebra of T. We write R for the (absolute) root system of T in G. We often write G for the group of F-points of G, etc. We will follow closely the exposition of Harish-Chandra's work given in [66]. Most of the proofs are just the same as the ones there and will therefore be omitted. (Instead of a proof, the reader will find the words "same as usual.") However, a couple of additional ingredients will be needed; these are simple adaptations of ideas in Sparling's article [100].

A.2 Orbital integrals OX for X e t

Let X e t. The centralizer of X in G is a connected reductive F-subgroup of G that we will denote by MX. (The reason for using the letter M is that this subgroup is a twisted Levi subgroup of G, i.e. an F-subgroup that becomes a Levi subgroup after extending scalars to an algebraic closure of F; however this fact is not actually needed below.) The set M of subgroups obtained in this way (as X varies in t) is finite.

The following notation will be useful. Let M e M. We write RM for the (absolute) root system of M (a subset of R). We write zM for the Lie algebra of the center of M; we then have

Zm = {X e t: a(X) = 0 Va e Rm}

For X e t we have MX = M if and only if

{a e R: a(X) = 0} = Rm

or, in other words, if and only if X lies in the open subset

zM = {X e zm : a(X) = 0 Va e R\Rm}

of zM. Obviously t is the disjoint union of the locally closed subsets z'M. For example we have z'G = Zg , while z'T is the set of regular elements in t.

We fix a Haar measure dg on G. In addition, for each M e M we fix a Haar measure dm on M. For instance one can use the canonical measures defined by Gross. In any case, for X e z'M we define the orbital integral OX by

Ox(f) := i f (g-1 Xg)dg/dm. (13.2)

Thus we now have a coherent definition of orbital integrals for all X e t. A.3 Preliminary definition of Shalika germs on g

There are finitely many nilpotent G-orbits O1, O2,. ..,Or in g. We write for the corresponding nilpotent orbital integrals. The distributions ¡i1,...,Mr are linearly independent.

Theorem 13.2 There exist functions T1, T2,..., Tr on t having the following property. For every f e CC°(g) there exists an open neighborhood Uf of 0 in t such that

Ox(f) = Y. l^i (f) • Y (X) (13.3)

for allX e U f. The germs about 0 e t of the functions T1,..., Tr are unique. We refer to r, as the provisional Shalika germ for the nilpotent orbit Oi.

Proof Same as usual. □

A Shalika germ is an equivalence class of functions on t. As we will see next, the homogeneity of Shalika germs makes it possible to single out one

particularly nice function r within its equivalence class. Once we have done this, r will from then on denote this function (whose germ about 0 is the old r).

A.4 Behavior under scaling

For f e Fx and f e CC°(g) we write fp for the function on g defined by

fp(X) := f (f X). (13.4)

Harish-Chandra proved that

lo (fa2) = |a|-dim O lo (f) (13.5)

for every nilpotent orbit O and a e F x. Moreover it is clear from (13.2) that

Ox (fp) = Opx (f) (13.6)

for all X e t and all f e Fx .

A.5 Partial homogeneity of our provisional Shalika germs ri

Let a e F x. Let Oi be one of our nilpotent orbits, let n be the corresponding nilpotent orbital integral, and let ri be the corresponding Shalika germ. Put di := dim Oi . We claim that

r (X) = laldi r (a2X), (13.7)

where the equality means equality of germs about 0 of functions on t.

Indeed, as in the proof of the Shalika germ expansion on G, pick a function fi e C™(q) such that

l j (fi) = Stj. (13.8)

Then r (X) is the germ about 0 of the function

X ^ Ox(fi) (13.9)

on t. In fact during the remainder of our discussion of provisional germs, we will use always use (13.9) as our choice for a specific function r having the right germ.

In view of the homogeneity of nilpotent orbital integrals established above, ia idi • (fi )a2 can also serve as fi, so that V;- (X) is also the germ about 0 of the function

on t. Comparing (13.9), (13.10), we see that the germs of H (X) and la ldi n (a2X) are equal, as desired.

A.6 Canonical Shalika germs

Let n be one of our germs. We are going to replace H by another function Pnew on t that has the same germ about 0 and is at the same time homogeneous.

Lemma 13.3 There is a unique function T;new on t which has the same germ about 0 as Ti and which satisfies (13.7) for all a e F x andallX e t.Moreover pnew is real-valued, translation invariant under the center of g, and invariant under conjugation by elements in the normalizer of T.

Proof Same as usual. □

From now on we replace the germs T; by the functions pnew, but we drop the superscript "new."

We also need a slight strengthening of the fact that T; is translation invariant under the center z of g. Let G' be the derived group of the algebraic group G. Then G (F) = G'(F) Z (F ),butfor F -points we have only that G' Z isanormal subgroup of finite index in G. We denote by D the finite group G' G' Z. Each G-orbit O in g = g' ©z decomposes as a finite union of G'-orbits O', permuted transitively by D. We normalize the invariant measures on the orbits in such a way that

For a nilpotent G-orbit O (respectively, nilpotent G'-orbit O') we denote by VG (respectively, V£') the corresponding Shalika germ on t (respectively, g' n t).

Lemma 13.4 Let X e t and decompose X as X' + Z with X' e g' n t and Z e z- Then

Proof Same as usual, but note that there is a typo in the proof of the corresponding result in [66]: the functions f, f' occurring in formula (17.8.9) of that article should have a subscript O.

X - Ox (|a|di • (ft a) = |a|di • Oa2 x (fi) (13.10)

(13.11)

VG (X) = X ro (X')■

(13.12)

A.7 Germ expansions about arbitrary central elements in g

We have been studying germ expansions about 0 e t. These involve orbital integrals for the nilpotent orbits 0;. Now we consider germ expansions about an arbitrary element Z in the center of g. These will involve orbital integrals /Z+Oi for the orbits Z + Oi, but will involve exactly the same germs H as before.

Theorem 13.5 Let Z be an element in the center of g. For every f e CC°(g) there exists an open neighborhood Uf of Z in t such that

Ox(f) = X XZ+Oi (f) • Y (X) (13.13)

for all X e Uf.

Proof Same as usual. □

A.8 Germ expansions about arbitrary semisimple elements in g

We are going to use Harish-Chandra's theory of semisimple descent in order to obtain germ expansions about an arbitrary element S e t. We fix such an element S and let H := GS denote the centralizer of S, a connected reductive subgroup of G.

Let Y1,...,Ys be a set of representatives for the nilpotent H-orbits in h. Let XS+Yj denote the orbital integral on g obtained by integration over the G-orbit of S+Y .NowT is also a maximal torus in H ,soforeach1 ^ i ^ s wecancon-sider the canonical Shalika germ rf for H, t and the nilpotent H-orbit of Y; .

Theorem 13.6 Let S, H be as above. For every f e CC°(g) there exists an open neighborhood U f of S in t such that

OX(f) = X XS+Yi (f) • rH(X) (13.14)

for all X e U f. Proof Same as usual.

A.9 Normalized orbital integrals and Shalika germs For X e t we put

Dg(X) = det(ad(X); g/mx)

(mX being the Lie algebra of the centralizer MX of X in G) and define the normalized orbital integral IX by

Ix = |Dg (X )|1/2 Ox .

When we use IX instead of OX, we need to use the normalized Shalika germs T i (X) := | Dg (X )|1/2T (X) instead of the usual Shalika germs.

Clearly Theorem 13.2 remains valid when OX, TX are replaced by IX, T; respectively. Now consider the germ expansion about an arbitrary element S e t. As usual put H := GS. There exists a neighborhood of S in t on which

lDg(X)|1/2 = |Dh(X)|1/2|det(ad(S); g/h)|1/2.

It then follows from Theorem 13.6 that

Ix(f) = |det(ad(S); g/h)|1/2^ i^s+y,(f) • TH(X) (13.15)

for all X in some sufficiently small neighborhood of S in t.

The homogeneity property (13.7) of the Shalika germs T; implies the following homogeneity property for the normalized Shalika germs Ti:

T(a2X) = |a|dim(Gxi)-dim(mx} • T(X) (13.16)

for all a e Fx and all X e t. Here we have chosen Xi e Oi and introduced its centralizer G Xi .

The next proposition will be needed when we use (13.16) in the proof of boundedness of normalized Shalika germs. It is a simple adaptation of ideas from Sparling's article [100]. To formulate the proposition we need a definition. Consider the action morphism G x g — g (given by (g, X) — gXg-1); we are now thinking of G and g as algebraic varieties over F. For M e M we consider the image VM C g of G x zM under this morphism. Obviously VM is an irreducible G-invariant subset of the variety g, so its Zariski closure VM is a G-invariant irreducible subvariety of g. We say that a nilpotent orbit O is relevant to M if O is contained in VM .

Proposition 13.7 Let M e M and let O be a nilpotent orbit in g. Then the following two statements hold.

(i) If O is relevant to M, then for Y e O we have dim GY ^ dim M, where GY denotes the centralizer ofY in G.

(ii) If O is not relevant to M, then the normalized Shalika germ TO vanishes identically on z'M.

Proof (1) Over VM we have the group scheme whose fiber at X e VM is the centralizer of X in G. At points in ZM this centralizer is M and at points of VM it is some conjugate of M. Since VM is dense in VM, we conclude from SGA 3, Tome I, Exp. VIB, Prop. 4.1 that dim GX ^ dim M for all X e VM .In particular this inequality holds when we take X to be Y e O c Vm .

(2) Let f e C^g and suppose that /O (f) = 0 for all nilpotent orbits O relevant to M. Then, as in the proof of the existence of Shalika germs, there exists an open neighborhood Uf of 0 in t such that OX (f) = 0 for all X e Uf n VM .In particular OX (f) = 0forallX e Uf nZM. Applying this observation to the functions fj used to produce our provisional Shalika germs, we conclude that if Oj is not relevant to M, then there is a neighborhood Uj of 0 in t such that the provisional Shalika germ T j vanishes on Uj n z'M. Looking back at how the true (homogeneous) Shalika germs were obtained from the provisional ones, we see that the true Shalika germ T j vanishes identically on %'M when O j is not relevant to M. □

A.10 Ti is a linear combination of functions TH in a neighborhood of S

Again let S e t and let H be its centralizer in G. Consider one of the normalized Shalika germs Ti for G. We are interested in the behavior of Ti in a small neighborhood of S in t.

Lemma 13.8 There exists a neighborhood V of S in t such that the restriction of Ti to V is a linear combination of restrictions of normalized Shalika germs for H.

Proof Same as usual. □

Corollary 13.9 Let M e M. Each normalized Shalika germ Ti is locally constant on z'M.

Proof Same as usual. □

A.11 Locally bounded functions

We are going to show that the normalized Shalika germs Ti are locally bounded functions on t. First let's recall what this means. Let f be a complex-valued function on a topological space X. We say that f is locally bounded on X if every point x e X has a neighborhood Ux such that f is bounded on Ux . When X is a locally compact Hausdorff space, f is locally bounded if and only f is bounded on every compact subset of X.

A.12 Local boundedness of normalized Shalika germs

Let Ti be one of our normalized Shalika germs on t. We are going to show that Ti is locally bounded as a function on t, slightly generalizing a result of Harish-Chandra.

Theorem 13.10 Every normalized Shalika germ Ti is locally bounded on t.

Proof Same as usual once one takes into account Proposition 13.7. □

As a consequence of the local boundedness of normalized Shalika germs, we obtain a slight generalization of another result of Harish-Chandra.

Theorem 13.11 Let f e C£°(g). Then the function X — IX (f) on t is bounded and compactly supported on t. Moreover, for each M e M this function is locally constant on z'M.

Proof Same as usual. □

Appendix B: By Raf Cluckers, Julia Gordon and Immanuel Halupczok

In this appendix we use the theory of motivic integration to control bounds for orbital integrals, normalized by the discriminant, as the place varies. In Appendix A, the bound for orbital integrals is proved for a fixed local field; here we show that this bound cannot exceed a power of the cardinality of the residue field, using the tools from model theory. We emphasize that the main result of Appendix A, namely, the fact that the orbital integrals are bounded, is used in our proof. More specifically, we prove Theorems 14.1, and 14.2 which are stronger versions of, respectively, Theorem 7.3 and Proposition 7.1 with eG = 1. We also prove the analogous statement for the function fields; moreover, we prove that the optimal exponents can, in some sense, be transferred between the function field and number field cases, see Theorem 14.7. We expect that the same methods could apply to weighted orbital integrals, provided that one had a statement similar to the Theorem 13.1 of Appendix A.

Let F be a number field with the ring of integers O. Let G be a connected reductive algebraic group defined over F, and g its Lie algebra. Let F = Fv be a completion of F. We denote the ring of integers of F by OF, the residue field by kF, and let qF = #kF be the cardinality of kF. For a semisimple element Y e G(F) and a test function f e C^°(G(F)), the orbital integral at y is denoted by OY (f). As in Appendix A,

DG(Y) = IT 11 - a(Y)lv

aeO a(Y)=1

where O is the root system of G.

We keep the set-up of Sects. 7.1 and 7.2; in particular, we first treat the case of a reductive group with a given root datum defined over a local field, and then derive the global statement from it. Thus, we start with a reductive group G defined over a local field F, and we assume that G is unramified. In order to get to this setting from the global set-up, we just have to assume that G = Gv where the place v is finite, and lies outside the set Ram(G).

Given an unramified reductive group G over a local field F as above, we recall the definition of the functions tG from Sect. 2.2. We have a Borel subgroup B = TU, and let A be the maximal F-split torus in T. As in Sect. 2.2, choose a smooth reductive model G for G corresponding to a hyperspecial point in the apartment of A, and let K = G(OF) be a maximal compact subgroup. For X e X * (A), tG is the characteristic function of the double coset KX (m) K.

We prove

Theorem 14.1 Let G be a connected reductive algebraic group over F, with T and Av as in Sect. 7.2. There exist constants aG and bG that depend only on the global model of G such that for all X e X *( Av) with ||X|| ^ k , for all but finitely many places v

Oy(TG)

< q%G +bGKDG(y)-1/2

for all semisimple elements y e G (Fv), where qv is the cardinality of the residue field of Fv.

In fact, we prove a stronger and more general statement, which does not require F to have characteristic zero. By an unramified root datum we mean a root datum of an unramified reductive group over a local field F, i.e. a quintuple £ = (X*, 4>, X*, 4>v, 6), where 6 is the action of the Frobenius element of Fur / F on the first four components of £.

Theorem 14.2 Consider an unramified root datum £. Then there exist constants M > 0, a£ and b£ that depend only on £, such that for each non-Archimedean local field F with residue characteristic at least M, the following holds. Let G be a connected reductive algebraic group over F with the root datum £. Let Abe a maximal F-split torus in G, and let tG be as above. Then for all X e X*(A) with ||X|| ^ k,

Oy(TG)

< qFa£ +b£KDG(y)-1/2

for all semisimple elements y e G(F).

The strategy of the proof is to use the theory of motivic integration developed by Cluckers and Loeser [29]. In [29], a class of functions called conS

structible motivic functions is defined. Here, in order to simplify the language, we are working directly with the specializations of constructible motivic functions, which we define below, and we call these "constructible functions". These functions are defined by means of formulas in a first-order language of logic, called Denef-Pas language, which we review below. The key benefit of using logic is that the formulas defining the functions are independent of the field of definition, hence this set-up is perfectly suited for proving a result that applies uniformly across almost all completions of a given number field. This method can be thought of as an extension of a geometric approach—"definable" is a less restrictive notion than "geometric", yet it provides a field-independent way of talking about orbital integrals.

The key to our proof is a general result which, roughly speaking, states that if a constructible function is bounded (which is known in our case thanks to Appendix A), then its upper bound cannot exceed a fixed power of the cardinality of the residue field (Theorem 14.6 below). In order to apply this result to orbital integrals, we need to show that they are, in some sense, constructible functions. More precisely, one would like to show that given a constructible test function f e C^°(G(F)), the function y — OY( f) is a constructible function of y, on the set of all semisimple elements. For regular semisimple elements, the Lie algebra version of this statement is essentially proved by Cluckers et al. [27]. For general elements X, the Lie algebra version of this statement with a particular normalization of the measure on the orbit is proved in [26]; however, the normalization of the measures used in [26] is not the same as the canonical normalization used in Appendix A above. For non-regular semisimple elements, we show here that the canonical measure differs from a motivic measure by a constant that can be bounded by a fixed power of the cardinality of the residue field, and consequently, obtain that given f, there exists a constructible function Hf and a constant c that depends only on the root datum of the group, such that | Hf (y)| ^ | OY( f )| ^ qc | Hf (y)|. Taking f to be the characteristic function of the maximal compact subgroup K in this argument, we obtain the special case of Theorem 14.2 with k = 0. The full statement of Theorem 14.2 is obtained by a similar argument that allows the test functions to vary in definable families.

Much of the preliminary and introductory material is quoted freely from [25-28,43], sometimes without mentioning these ubiquitous citations.

B.1 Denef-Pas language

The Denef-Pas language is a first order language of logic designed for working with valued fields. We start by defining two sublanguages of the language of Denef-Pas: the language of rings and Presburger language.

B.1.1 The language of rings

A formula in the first-order language of rings is any syntactically correct formula built out of the following symbols:

- constants '0', '1';

- binary functions 'x', '+';

- countably many symbols for variables x1,...,xn,... running over a ring;

- the following logical symbols: equality '=', parentheses '(',')', the quantifiers '3', 'V', and the logical operations conjunction 'a', negation '—', disjunction 'v'.

If a formula in the language of rings has n free (i.e. unquantified) variables then it defines a subset of Rn for any ring R. Note that quantifier-free formulas in the language of rings define constructible sets (in the sense of algebraic geometry).

B.1.2 Presburger language

A formula in Presburger's language is built out of variables running over Z, the logical symbols (as above) and symbols '+', '0', '1', and for each d = 2, 3, 4, ..., a symbol '=d' to denote the binary relation x = y (mod d). Note the absence of the symbol for multiplication.

Since multiplication is not allowed, sets defined by formulas in the Pres-burger language are in fact very basic, cf. [23] or [81]. For example, {(a, b) e Z2 | a = 1 mod 4; a < b +10} is a Presburger subset of Z2. Since quantifiers are never needed to describe Presburger sets, they all are of a similar, simple form.

B.1.3 Denef-Pas language

The formulas in Denef-Pas language have variables of three sorts: the valued field sort, the residue field sort, and the value group sort (in our setting, the value group is always assumed to be Z, so we will call this sort the Z-sort). Here is the list of symbols used to denote operations and binary relations in this language:

- In the valued field sort: the language of rings.

- In the residue field sort: the language of rings.

- In the Z-sort: the Presburger language.

- a symbol ord(•) for the valuation map from the nonzero elements of the valued field sort to the Z-sort, and ac( •) for the so-called angular component, which is a function from the valued field sort to the residue field sort (more about this function below).

On top of the symbols for the constants that are already present (like 0 and 1), we will add to the Denef-Pas language all elements of O[[t]] as extra symbols for constants in the valued field sort. We denote this language by LO.

Given a discretely valued field F that is an algebra over O, together with a choice of a ring homomorphism i : O — F and a choice of a uniformizer m of the valuation, one can interpret the formulas in LO by letting the variables range, respectively, over F, the residue field kF of F, and Z (which is the value group of F). The function symbols ord and ac are interpreted as follows. For x e F x, ord(x) denotes the valuation of x. If x is a unit (that is, ord(x) = 0), then ac(x) is the residue of x modulo m (thus, an element of the residue field). For a general x = 0 define ac(x) as ac(m -ord(x)x); thus, ac(x) is the first nonzero coefficient of the m -adic expansion of x. Finally we define ac(0) = 0. The elements from O are interpreted as elements of F by using i, the constant symbol t is interpreted as the uniformizer m, and thus, by the completeness of F, elements of O[[t]] can be naturally interpreted in F as well.

Definition 14.3 Let CO be the collection of all triples (F,i,m), where F is a non-Archimedean local field which allows at least one ring homomorphism from O to F, the map i : O — F is such a ring homomorphism, and m is a uniformizer for F. Let AO be the collection of those triples (F,i,m) in Co in which F has characteristic zero, and let BO be the collection of those triples (F, i, m) where F has positive characteristic.

Given an integer M, let CO,M be the collection of (F,i,m) in CO such that the residue field of F has characteristic larger than M, and similarly for AO,M and BO,M.

Since our results and proofs are independent of the choices of the map i and the uniformizer m, we will often just write F e CO, instead of naming the whole triple. For any F e CO, write OF for the valuation ring of F, kF for its residue field, and qF for the cardinality of kF.

In summary, an LO-formula f with n free valued-field variables, m free residue-field variables, and r free Z-variables defines naturally, for each F e CO, a subset of Fn x k^F x Zr by taking the set of all tuples where f is "true" (in the natural sense of first order logic, see e.g. [73]).

B.2 Definable sets and constructible functions

As mentioned in the introduction, to study dependence on p of various bounds we will need to have a field-independent notion of subsets of Fn x kmF x Zr for F e CO. To achieve this, we call a collection (XF)F of subsets XF c Fn x km x Zr, where F runs over CO, which come from an LO-formula f as explained at the end of Sect. B.1.3, a definable set. Thus, for us, a "definable set" is actually a collection of sets, namely one for each F e CO; in earlier work

on motivic integration, the term "specialization of a definable subassignment" was used for a similar notion. For an integer r > 0, Zr will often denote the definable set (XF) F such that XF = Zr for each F. More generally, for nonnegative integers n, m, r, the notation h[n, m, r] will stand for the definable set (Fn x jf x Zr)f.

For definable sets X and Y, a collection f = (fF )F of functions fF : XF ^ Yf for F e CO is called a definable function and denoted by f: X ^ Y if the collection of graphs of fF is a definable set.

Definable functions are the building blocks for constructible functions, which are defined as follows. For a definable set X, a collection f = (fF )F of functions fF : XF ^ C is called a constructible function if there exist integers N, N', and N", such that fF has the form, for x e XF, for all F e CO,

n in' \ /N" 1 \

fF(x) = XqF'F(x)#(p-F(x)) (nfrjF(x)j (n YZqEj.

where:

- ate with i = 1,..., N, I = 1,..., N" are negative integers;

- ai: X ^ Z with i = 1,... N, and frj: X ^ Z with i = 1 ..., N, j = 1,..., N' are Z-valued definable functions;

- Y; are definable sets such that YiF c kF x XF for some ri e Z, and pi: Y; ^ X is the coordinate projection.

The motivation for such a definition of a constructible function comes from the theory of integration: namely, constructible functions form a rich class of functions which is stable under integration with respect to parameters (as in Theorem 14.4 below). See [28,43] for details.

For each F in CO, let us put the Haar measure on F so that OF has measure 1, the counting measure on kF and on Z, and the product measure on Cartesian products. Thus, we get a natural measure on h[n, m, r]. Furthermore, any analytic subvariety of Fn, say, everywhere of equal dimension, together with an analytic volume form, carries a natural measure associated to the volume form, cf. [13].

The notion of a measure associated with a volume form carries over to the definable setting, roughly as follows. By the piecewise analytic nature of definable sets and definable functions, any definable subset X of h[n, m, r] can be broken into finitely many pieces Xi , such that Xi ( F) is a subset of V; x k'm x Zr for some F-analytic subvariety V; of Fn of the same dimension as X; (F), for each F with large residue characteristic. A definable form on h [n, 0, 0] in the affine coordinates x is just a finite sum of terms of the form f (x )dxi1 a • • • Adxid where f is a definable function with values in h [ 1, 0, 0] .If the functions f restrict to F -analytic functions on V; for each such f, and if the

form is a d-form where d is the dimension of Vi, then one can use the measure associated to this analytic volume form on Vi. This construction yields natural "motivic" measures on the definable set X, associated to definable differential forms, cf. also [26, §3.5.1]. Such a construction of measures associated with differential forms behaves well in the setting of motivic integrals because there exists a natural change of variables formula for motivic integrals, see §15 of [29]. In summary, the measures that arise from definable differential forms occur naturally in the context of motivic integration and we will call such measures "motivic" below. We refer to [29, §15] for the definition of the sheaf of definable differential forms on a definable set, and other details. We note that any algebraic volume form on a variety over OF, where F is a global field, is definable in this sense. Note, however, that in this appendix we have to deal with volume forms on orbits of elements of a group defined over a local field, and the resulting measures are not automatically motivic.

Let us recall one of the results of [25], the first part of which generalizes a result of [30], and which shows that the class of constructible functions is a natural class to work with for the purposes of integration.

Theorem 14.4 [25, Theorem 4.3.1] Let f be a constructible function on X xY for some definable sets X and Y. Then there exist a constructible function g on X and an integer M > 0 such that for each F e CO,M and for each x e XF one has

gF (x) = / fF (x, y), JyeYp

whenever the function YF — C: y — fF (x, y) lies in L 1(YF), where, say Yf C Fn x km x Zr.

Note that although the theorem is stated for the affine measure on Fn, it also holds for measures given by definable differential forms, by working with charts as is done in [29, §15].

Remark 14.5 In the literature on general motivic integration, one often uses a more abstract notion of "definable subassignments". Any such definable subassignment X specializes to the sets XF discussed here for all F e Co,M for some M, and any motivic integral over X specializes to the corresponding integrals over XF .In this paper it is sufficient and more convenient to work with the above notion of definable sets ( XF) F directly.

Let us finally fix our terminology about "families of definable sets" and "families of constructible functions". A family of definable sets Xa indexed by a parameter a e A is a definable subset X of Y x A for some definable sets Y and A, equipped with the canonical projection pA: X — A,

and the family members are p-x(a) = Xa for a e A. Similarly, a family of constructible (respectively, definable) functions fa on the family Xa is a constructible (respectively, definable) function on X c Y x A. Whenever we call a specific function f: Xf0 c F^ x km x Zr ^ C (for a specific field F0) constructible, we mean that it appears naturally as fF0 for a constructible function (fF) f for which uniformity in F is clear from the context as soon as the residue field characteristic is large enough; we use a similar convention for calling a specific function definable, and so on.

Finally, we will occasionally need to take roots of q (in order to take the square root of the absolute value of the discriminant, for example). We adopt the same convention as in [26, §B.3.1], and call any expression of the form 1 f

HqrF , or a finite sum of such expressions, a motivic function, where f is a Z-valued definable function, and H is a motivic function in the usual sense defined above. All the results about motivic functions generalize to this setting by splitting the domain into finitely many pieces according to (f mod r). We note also that the boundedness results from the next section for such functions reduce to the same results without fractional powers by considering the r-th power.

B.3 Boundedness of constructible functions

The following two theorems are the main results of this section.

Theorem 14.6 Let H be a constructible motivic function on W x Zn, where W is a definable set. Then there exist integers a, b and M such that for all F e CO,M the following holds.

If there exists a (set-theoretical, and not necessarily uniform in F) function aF: Zn ^ R such that

IHf(w,X)|r < aF(X) on Wf x Zn,

then one actually has

|Hf(w, X)|R < qaF+Hn on Wf x Zn,

where ||X|| = X¿I=1 Xi, and where | • |R is the usual absolute value on R.

We observe that in the case with n = 0, the theorem yields that if a constructible function H on W is such that HF is bounded on WF for each F e CO,M, then the bound for | HF |R can be taken to be qF uniformly in F with large residue characteristics, for some a > 0.

The following statement allows one to transfer bounds, which are known for local fields of characteristic zero, to local fields of positive characteristic, and vice versa.

Theorem 14.7 Let H be a constructible motivic function on W x Zn, where W is a definable set, and let a and b be integers. Then there exists M such that, for any F e Co,M, whether the statement

Hf(w, k) < gF+b'WI for all (w, k) e Wf x Zn (14.1)

holds or not, only depends on the isomorphism class of the residue field of F.

Informally speaking, the idea of the proof is to first eliminate all the valued-field variables, possibly at the cost of introducing more residue-field and Z-valued variables. This step is summarized in Lemma 14.8 below, whose proof relies on the powerful cell decomposition theorem for definable sets in Denef-Pas language. Once we have a constructible function that depends only on the residue-field and value-group variables, we note that the residue-field variables can only play a very minor role in the matters of boundedness (the so-called "orthogonality of sorts" in Denef-Pas language referred to below). Finally, the question is reduced to the study of Presburger constructible functions of several Z-variables, which are similar to constructible functions as defined above in Sect. B.2, but without the factors #(p~[F(x)), see [25]. Roughly, Presburger constructible functions in x e Zr are sums of products of piecewise linear functions in x and of powers of qF, where the power also depends piecewise linearly on x .If such a function is bounded, then it is a sum of bounded terms as above, after removing possible redundancy in the sum. Each single term in x can then easily be bounded, by a power of qF that depends linearly on x. Since the number of terms is bounded, one obtains an upper bound of the right form. The reduction to single terms instead of their sum is made precise via the Parametric Rectilinearization (see Theorem 2.1.9 of [25]) and Lemma 2.1.8 of [25]. In summary, the main tools used to obtain these rather strong results with seeming ease are the cell decomposition theorem and the understanding of Presburger constructible functions. Now we proceed with the detailed proof.

Proposition 14.8 Let H be a constructible function onW x B for some definable sets W and B. Then there exist a definable function f: W x B — h [0, m, r ]x Bforsomem > 0 andr > 0, which makes a commutative diagram with both projections to B, and a constructible function G on h[0, m, r ]x B such that, for some M and all F in Co,M, the function HF equals the function GF o fF, and such that GF vanishes outside the range of fF.

Proof Let us write W c h [n, a, b] for some integers n, a and b. It is enough to prove the lemma when n = 1 by a finite recursion argument. We are done since the case n = 1 follows from the Cell Decomposition Theorem, in the version of Theorem 7.2.1 from [29]. □

Proof (Proof of Theorem 14.6) Let us first consider the specific case that, for each F e CO,M for some M, the set WF is a subset of Zr for some r > 0 and that HF is of the specific form, mapping x e WF x Zn to

XsiF^FF(x} (EI PiJF(x )

i=1 \j=1

for some real numbers siF possibly depending on F but not on x, and some definable functions ai: X ^ Z and fiij: X ^ Z. Let us moreover assume that W as well as the graphs of the ai and fiij are already definable in the Presburger language (which is a sublanguage of the Denef-Pas language). Let us finally assume that there exists a0 > 0 such that < qF° for each i and F. Let us call the specific situation with all these assumptions case (1). This case (1) reduces to the case that the ai and fiij are restrictions of Z-linear functions and that W = As x N for some I > 0 and some finite set As depending on s e Zn by Theorem 2.1.9 of [25] applied to X = S x W with S = Zn in the notation of that theorem. If As is a singleton, then the result follows from Lemma 2.1.8 of [25]. For As with at least two elements, one replaces HF by the sum of (HF + 1)2 over the elements of As and the proof is completed by Theorem 14.4 and induction on r.

The more general case where W c h[0, m, r ] for some m > 0 and some r > 0 can be reduced to case (1) by the orthogonality between the residue field sort and the value group sort. Concretely, the following form of orthogonality, see [104], is used. For any definable set A c h[0, m, r] there exist M > 0 and finitely many definable sets Bi and C; such that Bi c h[0, m, 0] and C; c h [0, 0, r] for each i, and AF = |Ji BiF x CiF for each F e CO,M, see (3.5) and (3.7) of [104]. It is this form of orthogonality that is applied to all the Denef-Pas formulas that are used to build up H (recall that constructible functions are built up from definable functions, and hence, involve finitely many formulas).

For the general case of the theorem, let us choose f: W xZn ^ h[0, m, r ]x Zn and G with the properties as in Lemma 14.8 with B = Zn. For G instead of H and h [0, m, r] instead of W, we know that the theorem holds by the above discussion. But then the theorem for H follows. Indeed, by Proposition 14.8, the set Hf (Wf x {X}) U {0} equals (as subset of R) the set Gf(k% x Zr x

{X }) U {0} for each X e Zn and each F in COtM/ for some M

Proof (Proof of Theorem 14.7) If W c h[0, m, r ] for some m > 0 and some r > 0, then, for some M, the function

HF: WF x Zn ^ C

will depend on F only via the two-sorted structure on (kF, Z) coming from restricting the Denef-Pas language LO to the sorts (kF, Z) (i.e., leaving out the ring language on the valued field sort and the symbols ord and ac).

Hence, for W c h[0, m, r] the theorem follows. Now the general case follows from the case W c h[0, m, r] by Proposition 14.8. □

B.4 Root data and reductive groups B.4.1 Split reductive groups

We start out by following [27] in the treatment of the root data and definability of the group G and its Lie algebra g. Split reductive groups G are classified by the root data ^ = (X*, 4>, X$v) consisting of the character group of a split maximal torus T in G, the set of roots, the cocharacter group, and the set of coroots. The set of possible root data of this form (which we will refer to as absolute root data) is completely field-independent. Given a root datum ^, the group G(F) is a definable subset of GLn(F), given as the image of a definable embedding S: G — GLn, defined over Z[1/R] for some large enough R (see Sect. 7.2 of the main article; we note also that in [27], such an embedding is denoted by pD).

In order to show that general reductive groups are definable, we will use the fact that every reductive group splits over the separable closure of F, and the F-forms of a group are in one-to-one correspondence with the Galois cohomology set H 1(F, Aut(G)) (see e.g. [101, §16.4.3]).

We start by giving a construction of finite separable field extensions in Denef-Pas language.

B.4.2 Field extensions

Let [T] be an isomorphism class of the Galois group of a finite field extension. We can think of a representative of [T] explicitly as a finite group determined by its multiplication table. Given a non-Archimedean local field F, we would like to realize all field extensions of F with Galois group in the isomorphism class [T] as elements of a family of definable sets (with finitely many parameters coming from F). Let m be the order of T. Let b = (bo ,...,bm-1) e Fm. The set of tuples b such that the polynomial Pb(x) = xm + bm-1 xm-1 + ••• + b0 is irreducible and separable, is definable. As in [27, §3.1], one can identify the field extension Fb = F[x]/(Pb (x)) with Fm. Further, the condition that the field extension Fb / F is Galois is definable. Indeed, it is given by the requirement that Pb is irreducible over F, the degree of Fb over F equals m, and there exist m distinct roots of Pb (x) in Fb. Note that the latter condition is expressible in Denef-Pas

language using b as parameters, and an existential quantifier. Similarly to [27], we treat the elements of the Galois group Gal(Fb/F) as m x m-matrices of variables ranging over F. More precisely, we introduce m x m -matrices a1,...,am of variables ranging over F, and impose the condition that a1,...,am are distinct automorphisms of Fb over F, and there exists a bijection {a1,...,am} — T which is a group isomorphism. Finally, let

S[T] C Fm+m be the definable set of tuples (b,a1,..., am) satisfying the conditions defined above. Note that every Galois extension of F with the Galois group of the isomorphism class [T] will appear as a fibre of S[T] over h [m, 0, 0] several times, since a1,...am are not unique for each isomorphism type.

B.4.3 General connected reductive groups

Let ^ be an absolute root datum as in Sect. B.4.1 above, and let G be the corresponding split group (so that we can think of G as a definable set). The goal is to construct the sets G(F) for all connected reductive algebraic groups G with absolute root datum ^ as members in a family of definable sets GzF, indexed by a parameter z which, loosely speaking, encodes the information about the the cocycle Gal(Fsep/F) — Aut (G)(Fsep). More precisely, for every parameter s = (b,a1,...,am) e S[T] as above, we consider the groups G with the absolute root datum ^ that split over the extension Fb corresponding to the parameter b (if such groups exist). Such groups are in one-to-one correspondence with the elements of the set H1 (Gal(Fb/F), Aut(G)(Fb)). Following the approach of [27, §5.1], we work with individual cocycles rather than cohomology classes. First, observe that the family of sets Z 1(Gal(Fb/F), Aut(G)(Fb)) of such cocycles is a family of definable sets, indexed by s e S[T]. This follows from the fact that G is definable: indeed, then the group Aut(G)(Fb) is definable as well, and we have Gal(Fb/F) ~ [a1,...am}, and the cocycle condition is, clearly, definable.

Definition 14.9 We denote by Z[T] the definable set Z 1(T, Aut(G)(Fb)) equipped with the projection to the set S[T].

Let us now recall the construction of the group Gz (F) corresponding to the cocycle z. By definition, Gz (F) is the set of fixed points in G(Fb) under the action of Gal(Fb/F) ~ {a1,... am} given by: a ■ g = z(a)(ag), where g e G(Fb), a e Gal(Fb/F), and the action ag is the standard action of the Galois group, where a acts on the coordinates of g. Such a fixed point set is definable (with parameters from Z[T]), since a1,...,am are interpreted as matrices of variables with entries in F, according to Sect. B.4.2.

B.4.4 Unramified groups

In the case G is unramified over F, i.e. when it is quasi-split and splits over an unramified extension of F, one can think of G (F) as the fixed-point set of the action of the Frobenius element, which substantially simplifies the above construction, see [27, §4.2] for detail. Unramified reductive groups are determined by the root data £ = 6), where ^ is an absolute root datum as in Sect. B.4.1, and 6 is the action of the Frobenius automorphism on ^.

Remark 14.10 The reason we are including general reductive groups here even though we can, and will, assume that G is unramified over F, is that we have to deal with the connected centralizers of semisimple elements of G(F), and these can be quite general reductive groups.

When we start with a reductive group G over a global field F, outside of the set of places Ram(G), the group G xF Fv over Fv is unramified and there are finitely many possibilities for its root datum, as described in Sect. 5.2 of the main article. We recall the notation: the set of finite places v where G xF Fv is unramified is partitioned into the disjoint union of sets V (6), 6 e C(T1) (see Sect. 5.2 for the definitions). Accordingly, for every conjugacy class [6] e C(T1), we have a definable set, which we denote by G\o], such that G[e]Fv = G(Fv) for all v e V(6).

We emphasize that G [6]F, by construction, is a definable subset of GLn (F^) for a suitable parameter b, as in [27, §4.1].

B.5 Orbital integrals

Here we prove the main technical result - namely, that the orbital integrals are bounded on the both sides by constructible functions. Throughout this section, we are assuming that we are given an unramified root datum £ = 6). For every local field F of sufficiently large residue characteristic, it defines an unramified reductive group G, and also gives rise to a definable set G]F = G(F), as in Sect. B.4.4 above. Note that we are not assuming that F has characteristic zero.

B.5.1 Two lemmas

We start with two easy technical remarks.

Lemma 14.11 Let £ be an unramified root datum as above, F—a local field of sufficiently large residue characteristic, and G - the corresponding reductive group over F defined by the root datum £. Then the set of semisimple elements in G (F) is definable.

We will denote this definable set by GsFs.

Proof The proof is, in fact, contained in the proof of [27, Lemma 7.1.1]. Indeed, the lemma follows from the fact that existence of a basis of eigenvectors is a definable condition: we can write down the conditions stating that there exists a degree n! extension over which there exists a basis of eigenvectors for an element g e G (F) c GLn (F^) for a suitable prameter b. □

Next, we show that the functions v^ (see Sect. 2.2) forming the basis of the spherical Hecke algebra are constructible, and depend on in a definable way.

Lemma 14.12 Let G be an unramified reductive group with the root datum £ as above. Then there exists M > 0 (depending only on £) and a definable family of constructible functions T , such that for each F in Co,M one has that

= TX ,f ■

Proof For unramified groups, it is proved in [26] that the hyperspecial maximal compact subgroup K is definable. One can identify the parameter X with an r-tuple of integers (X1,..., Xr), where r is the rank of the maximal split torus in G. We can fix an isomorphism xA : A ^ (Gm)r defined over Z. For a e A, let 4*X(a) be the formula stating that there exists a tuple (t1,...tr) e (F x )r with ord(t;-) = X i for i = 1,..., r, such that xA(a) = (t1,..., tr). Then the double coset K K is defined by the condition on g:

3k1, k2 e K, a e A such that g = k1ak2, ^X(a) = 'true'.

Therefore, we can take T ,F to be the characteristic function of this double coset. □

B.5.2 The measures

Recall the normalization of the measures used to define the orbital integrals in the main article and in Appendix A.

Let y e G (F) be a semisimple element. Then IY (the connected component of the centralizer of y ) is a connected reductive group, and has a canonical measure d«can defined by Gross [47, §4]. The G-invariant measure on the orbit

d «can

Oy is defined as the quotient measure ¿«ran of the canonical measure d

on G by the canonical measure on Iy . This is the measure that appears in the statement of the main theorem. However, we do not yet know that this measure is "motivic" in general. The difficulty comes from the canonical measure on Iy itself in the case y is ramified. We point out that it is explained in [27,

§7.1] for split groups (and stated for unramified groups), that the canonical measure ddefined by Gross comes from a definable differential form, and therefore fits into the framework of motivic integration by the construction of [29, §8]. The same statement for ramified groups is still open. For now, we prove a technical lemma that allow us to circumvent this difficulty. Namely, we prove that there exists a motivic measure on the orbit, and that it differs from the canonical measure by a constant bounded on both sides by fixed powers of q.

Let M be a connected reductive group over F that splits over a tamely ramified extension. Let F1 be a finite Galois extension over which M splits, and let T = Gal(F1 /F). Let x be a special point in the building of M over F, and let M(F)x be the corresponding maximal compact subgroup of M(F). By definition of the canonical measure, ^M"(M(F)x) = 1. Our difficulty is that it is presently not known whether M(F)x is definable, except in the case when the group M is unramified over F. For our current purposes, a weaker statement will be sufficient.

In Sect. B.4.3 above, we have constructed M(F) as an element of a family of definable sets (using parameters in Z[T], with M in place of G), by taking the set of T-fixed points of M( F1), under the action determined by a cocycle z. It follows from [80] that M(F)x c M(Fx)x n M(F), see [1, Lemma 2.1.2] for the statement precisely in this form. Let M1 = M(F1 )x n M(F). Then the subgroup M1 is definable, since M(F1)x is definable because M is split over F1 (see [26]).

Definition 14.13 We denote by iM the index [M1: M(F)x].

The proof of the next crucial lemma was provided by Sug Woo Shin. Note that this is the only place where we need to assume that the extension F1 is tamely ramified. We observe also that a much more precise bound (which we do not need for our present purposes) could have been obtained using the results of Kushnirsky [69].

Lemma 14.14 With the notation as above, there exists a constant c depending only on the root datum of G such that

iM = [M1: M(F)x ] < qc

when F e CO and M runs over all connected centralizers of semisimple elements of G(F).

Proof Let M2 = M(F1)x,0+ n M(F) = M(F)x,0+, where the equality holds by Remark 2.2.2 of [1] (note that the field is not assumed to have characteristic zero in [1]). We have M2 c M(F)x c M1, so [M1: M(F)x] < [M1: M2]. Let Mx be the maximal reductive quotient of the reduction mod p of the OF -group scheme associated to the parahoric subgroup M(F)x by Bruhat-Tits,

see [77, §3.2] (where the group is denoted by G and the reductive quotient -by M). Then it follows from [77, §3.2] that M1 /M2 can be identified with the set of kF1 -points of Mx, where kF1 is the residue field of F1, and thus we get iM ^ #Mx(kF1). Since the dimension of Mx is at most the dimension of G, there is a bound on #Mx (kF1) given by Steinberg's formula (see [47, §3]); then we carry out the same estimate as done for the numerator in the Eq. (7.11) in the main article, to obtain

#Mx(kF1) < qfGqr1G(dg+1},

where rG and dG stand for rank and dimension of G, respectively, and q1 is the cardinality of kF1 . Finally, since the degree of the extension [ F1 : F] is bounded by a universal constant, we obtain the desired result. □

Now we can define a motivic measure on the orbit of y .As above, IY is the set of F-points of a connected reductive algebraic group, which we will denote by M. Let T be the Galois group of the finite field extension that splits M. Then M(F) = IY arises in a family of definable sets (with parameters in Z[T]) constructed in Sect. B.4.3 above. There exists a motivic measure on M(F) (which uses the cocycle z as a parameter, so we will denote it by djf1), constructed in [26, §3.5.2] (see also [44, §2.3]), and if M is unramified over F, this measure coincides with the canonical measure d¡M". Consider the

du,can

quotient measure ^mr on the orbit of y. Since G is unramified, this is a quotient of two motivic measures.

Recall the definable open compact subgroup M1 of M(F) = IY constructed above Definition 14.13. Let

d jc(an

d := voVot (M1) , (14.2)

and let Omot (f) be the orbital integral with respect to this measure. We will show in Lemma 14.15 below that this is a "motivic distribution" on C^°(G) in a precise sense.

For now, let us estimate the factor by which this distribution differs from the orbital integral with respect to the canonical measure. We have:

d ¡jf™ _ d ix™ d ¡Ffot djcan = d¡mot d|can '

j„mot

where -Aan is a constant, namely, the factor by which the Haar measure d jif01

a ¡iy i

that we defined on M(F) differs from the canonical measure on M(F). Since

by definition, the volume of the compact subgroup M(F)x with respect to the canonical measure is 1, we have that

= vold«m ot (M( F )x) = vold«mot (M1)/Im ,

where M1 and iM are as in Definition 14.13 above. Combining this with (14.2), we get:

diG 1 d«mot vold«m0t (M1) d«mot

d «7; = vold«m ot (M1) d « Iy\G iM = iMd «Iy\G'

Now we are ready to prove our main theorem. B.6 Proof of the main theorem B.6.1 Proof of Theorem 14.2

Let y e G F and let M = Iy be the identity component of the centralizer of Y, as above. We assume that the residue characteristic of F is sufficiently large so that IY is automatically tamely ramified. As above, f ^ Ojmot (f) denotes the distribution on C^°(G(F)) defined as the orbital integral with respect to the measure d«f°\G on the orbit of y.

Let us break up the definable set Gss into finitely many pieces according to the isomorphism class of the centralizer of y (see Appendix A). Fix a Galois group r, and suppose M is an algebraic group that splits over an extension F1 with Gal(F1/F) ~ T. Let ZM] be the definable set of Definition 14.9 with M in place of G. Let z e ZM] be a cocycle corresponding to M. We observe that the set of elements y such that IY is isomorphic to M, is definable, using b,o1,...,om and z as parameters (we are using the notation of Sect. B.4.3). For brevity, we denote this definable set by GMM [precisely, we should think of it as an element in a family of definable sets indexed by b,o1,...,om, z as above; in particular, we will denote by z(y) the cocycle that gave rise to M].

The following easy lemma amounts to the statement that the quotient of two motivic measures gives a motivic distribution, up to multiplication by a motivic constant (i.e., volume of a fixed definable set).

Lemma 14.15 Let {fs}se^ be a family of constructible test functions on G indexed by some definable set S.

Then there exists a motivic function HM on GMM x S

such that for all fields F of sufficiently large residue characteristic,

/ fs(Ad(g-1 )Y) d= HM(Y, s),

Jly \G(F)

forY e GM(F).

Proof By definition (which we quote from [66, §2.4]), the quotient measure

d «can

d^ is characterized by the identity

C (■ (■ d «can

f(g)d«Gn(g) = f(hg)d«mot(V^mk(g) (14.4)

JG(F) Jly \G(F )J Iy z d «m

for all f e CC°(G). We recall that this identity characterizes the quotient measure because the map a : C^°(G) — C£°(IY\G) defined by

f — (g — h f (hg)d«m0t(h))

is surjective. We observe that as we think of the measures as linear functionals on the spaces C^°(G), C^°(IY\G), etc., we can in fact replace these spaces with their respective subspaces consisting of constructible test functions. This follows form the fact that one can construct a family of definable balls, such that the space spanned by the characteristic functions of these balls is dense in the space C^°(G), and therefore constructible test functions still distinguish between continuous distributions. We refer to [24, §3] for details of such an argument.

Using the definable open compact subgroup M1 of I Y, we obtain that for every constructible function f e C£°(Iy\G), there exists a constructible function f e C£°(G) such that a(f ) = vold^mot(Mi) f. Thus, wejcan construct a family of constructible test functions fs such that a(fs)(g) = vol^(Mi) fs(Ad(g-1)y). Multiplying both sides of (14.4) by vol^mot(Mi), we obtain

d tf™

vol^mot ( Mi) fs (Ad (g~ )Y) ^ (g)

a( fs ^T^t = / fs (g)d № (g)- (14.5)

hY\g d «m

hy\^—Jd«mot Mf)

The left-hand side of (14.5) equals Jj \G fs(Ad(g-1)y)d«mo\G by definition; the right-hand side is a motivic function of all the parameters involved (i.e., of y, on which it depends directly and also via z(y), and of s) by the main theorem on motivic integrals, [29, Theorem 10.1.1] (briefly restated above as Theorem 14.4), since G is assumed to be unramified over F, and the canonical measure on G is motivic. □

Since the test functions t g form a definable family of constructible functions

by Lemma 14.12, the above lemma can be applied to this family, and yields

the existence of a constructible function HM (y, X ) on G M, such that

Hm (y,x) = oymot (tg )

for every у e GMM, X e Zn.

Therefore, by the relation (14.3), we have:

(tg) = or (tg) = 1

V / im v ' Im

= — hm (y, X ).

(14.6)

By Lemma 14.14, we have

Oy(tG) <

hM(y,x )

Oy(tG )

We observe that DG(y) is a constructible function of y, and by our convention on fractional powers of q, so is DG(y)1/2.

By the Theorem A.l, the function Oy(tg)DG(y)1/2 is bounded for every X. Therefore, the constructible function hM(y, X)DG(y)1/2 is bounded for every X, and now our Theorem 14.2 follows from Theorem 14.6.

B.6.2 Proof of Theorem 14.1

As discussed in Sect. 5.2 of the main article, the set of all unramified finite places is partitioned into finitely many families according to the root datum of the group G xF F„. Applying Theorem 14.2 to all these families and taking the maximum of the aG and bG values, we obtain Theorem 14.1.

Remark 14.16 Though our method sheds no light on the optimal values of aG and bG, Theorem 14.7 allows to transfer these values between positive characteristic and characteristic zero: namely, if, for example, some values aG and bG were obtained in the function fields case by geometric methods, Theorem 14.7 would immediately imply that the same values work for characteristic zero fields of sufficiently large residue characteristic. We also note that for good orbital integrals, it should be possible to get a bound on aG in terms of the dimension of G, using [35].

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