Proc. London Math. Soc. (3) 109 (2014) 553-595 © 2014 London Mathematical Society

doi:10.1112/plms/pdu011

Outer unipotent classes in automorphism groups of simple algebraic groups

Ross Lawther, Martin W. Liebeck and Gary M. Seitz Abstract

We study the unipotent elements of disconnected algebraic groups of the form G(t}, where G is a simple algebraic group in characteristic p possessing a graph automorphism t of order p. We classify the unipotent classes in the coset Gt and determine the corresponding centralizers, showing that these bear a close relation to classes in a certain natural connected overgroup of G(t}. We also obtain a formula for the total number of outer unipotent elements in the finite group Gy (t}, where 7 is a Frobenius morphism, analogous to the well-known Steinberg formula for the number of inner unipotent elements.

1. Introduction

Let G be a simple algebraic group over an algebraically closed field of characteristic p. There is a substantial literature concerning the unipotent classes of G and its finite analogues; for example, precise and detailed information on centralizers of representatives of these classes can be found in [11]. If the pair (G,p) is (Al, 2), (Dl, 2), (E6, 2) or (D4, 3) (with G being either simply connected or adjoint in the last case), there is a graph automorphism t of G of order p, and the coset Gt contains elements of order a power of p, which can be regarded as unipotent elements. Indeed, if we take G to be simply connected, then G(t) is contained in a larger simple algebraic group (respectively, Cl+i, Bl, E7 or F4), and the p-elements in Gt are unipotent elements of the larger group.

This paper is a contribution to understanding the outer unipotent elements in the disconnected group G(t) and corresponding finite analogues GY (t) for 7 a Frobenius morphism. At the outset, it is not evident that there are only finitely many conjugacy classes of such elements under the action of G. This assertion can be found in [16], and an elementary proof based on the finiteness of the number of unipotent classes in G is given by Fulman and Guralnick in [6, 2.6]. In Section 3, we shall use the method of [6] along with information on unipotent classes in G to determine the precise number of outer unipotent classes in the finite groups and then use this information to show that there are only finitely many outer unipotent classes in G(t).

There is also an elegant formula (see Theorem 1.1) giving the total number of outer unipotent elements in the extended finite groups, which is in the spirit of Steinberg's well-known formula for the number of inner unipotent elements (see [4, Theorem 6.6.1]).

A main goal of the paper is to obtain information on the conjugacy classes and precise centralizers of the outer unipotent elements. In addition, we want to explicitly relate the outer unipotent classes of Gt to the usual unipotent classes of the simple overgroup indicated above.

For (G,p) = (Di, 2), we can view the group G(t) as the full orthogonal group, and information on classes and centralizers, in both the algebraic and finite orthogonal groups, is given in [11, Chapters 6 and 7]. In this paper, we therefore mainly focus on the remaining

Received 4 June 2013; published online 25 March 2014.

2010 Mathematics Subject Classification 20G07 (primary), 20E45 (secondary).

The research of the second and third authors was supported by an EPSRC grant. We also thank Gunter Malle who suggested a possible approach to Theorem 1.1 using a result of Digne-Michel in [5, 3.18].

cases where (G,p) = (Al, 2), (Ee, 2) or (D4, 3). For the algebraic groups this was first taken up by Spaltenstein in [16] where he obtains information on conjugacy classes and dimensions of centralizers. For the finite groups Efs(q).2, D\(q).3 and Ael(q).2 with l ^ 5, information on outer classes and centralizer orders is given by Malle in [12, 13]. However, none of these papers determines the precise centralizers of the outer unipotent elements, in particular the reductive part of the centralizers.

Our aim here is to precisely identify the conjugacy classes of such elements in the algebraic groups, to determine centralizers of representatives of these classes, and to obtain similar information for the corresponding finite groups of Lie type. We achieve this in Theorems 1.2-1.5. All that we use from the literature above is Spaltenstein's description of classes and centralizer dimensions in the algebraic group Ai.2; however, we interpret this information within Cl+1, where it is quite natural and appears very much like corresponding results for the classical groups in characteristic 2 as developed in [11].

In the remainder of this introductory section we first set out a hypothesis which will apply throughout this paper and introduce some standard notation; we then state our main results. We begin with our general result on the number of outer unipotent elements in the finite groups; we then give our detailed results on conjugacy classes and centralizers separately for the Al case and for the E6 and D4 cases. Finally, we describe the structure of the rest of the paper.

1.1. Notation

Let G be a simple algebraic group over an algebraically closed field K of characteristic p; throughout this paper we shall assume that

(G,p) = (Ai, 2), (Di, 2), (Ee, 2) or (D4,3),

with G being either simply connected or adjoint in the last case. Let H be a simply connected simple algebraic group over K, of type Cl+1, Bl, E7 or F4, respectively.

Fix a maximal torus TG of G and take root subgroups with respect to TG; let E denote the root system of G, with simple system n = {a1,..., ar}, where r is the rank of G, and corresponding positive system E+. For 3 G E, let Xp denote the corresponding root subgroup, and xp : K ^ Xp be an isomorphism of algebraic groups, so that Xp = {xp(c) : c G K}; we assume the maps xp are chosen such that the Chevalley commutator relations hold. Let U = ripe£+ Xp. To simplify notation write up = xp (1) for 3 G E.

For j = 1,.. .,r let Sj = xaj (1)x_aj ( —1)xaj (1) be the standard representative in Nq(Tq) of the Weyl group reflection in the jth simple root. We shall write roots as linear combinations of simple roots, with the coefficients arranged in the form of the Dynkin diagram; thus, for example, if G = Ee or D4, the highest root in E is denoted 12221 or 121, respectively.

Given a group A and an automorphism 0 of A, write Ag = {a G A : ag = a} for the group of fixed points. Let t be a graph automorphism of G of order p which permutes the elements of n; we may assume that xaj (c)T = xajT (c) for all j and all c G K (the assumption on the isogeny type in the case (G,p) = (D4, 3) is required to ensure t exists). Then if G is simply connected, we may regard G(t) as a subgroup of H. Elements of the coset Gt which have order a power of p will be called outer unipotent elements.

Let q be a power of p, and a denote the q-field morphism of G satisfying xp(c)a = xp(cq) for all 3 G E and c G K; let 7 denote either a or aT. Then 7 is a Frobenius morphism of G. For example, if G = E6 is adjoint we have GY = Inndiag(Eg (q)), with e =1 or —1 according as y = a or aT (where Eg(q) = Ee(q) and E-1(q) = 2E6(q)). Write GYT = (GY)T.

Given a group A and an element a of A, denote by aA the conjugacy class in A containing a; our notation for unipotent classes is taken from [11], which extends the Bala-Carter labelling to cover all characteristics. If A is an algebraic group, Ru(A) denotes its unipotent radical, L(A)

denotes its Lie algebra, and VA(Aj) denotes the restricted irreducible A-module with high weight equal to the ith fundamental weight. Write Ud to denote a connected d-dimensional unipotent group, Td to denote a d-dimensional torus and Zd to denote the cyclic group of order d.

We conclude this subsection by observing that the proof of [4, Proposition 5.1.1] generalizes to the situation here to show that the quotient map G(t) ^ G(t)/Z(G) restricts to a bijective morphism between the varieties of outer unipotent elements of the two groups, which induces a bijection between the outer unipotent classes; moreover, Cz(g)(t) = 1, which means that if v € Gt is an outer unipotent element, then the quotient map induces an isomorphism between CG(v) and Cg/z(g)(vZ(G)). It follows that results obtained in G on outer unipotent classes and corresponding centralizers apply equally to any group isogenous to G (provided it admits t in the case (G,p) = (D4, 3)); it therefore suffices to treat one group in each isogeny class.

1.2. The number of outer unipotent elements

Assume (G,p) is any of the possibilities listed at the start of Subsection 1.1. Recall that Steinberg proved that the number of unipotent elements in GY (for any connected reductive group G and Frobenius morphism 7) is equal to (|G7|p)2 (see [4, Theorem 6.6.1]). The following result extends this formula to the situation treated here.

Theorem 1.1. Assume (G,p) = (A, 2), (A, 2), (E6,2) or (D4, 3). The number of p-elements in the coset GYt is equal to

GTr (|gy't ip)2-

Our proof of Theorem 1.1 shows that the conclusion also holds if we replace GY by Op (GY), and so holds in particular when this is a finite simple group.

1.3. The Al case

Assume (G,p) = (Al, 2); write n = l + 1 and take G to be simply connected, so that G = SLn(K). We can regard G(t) as a subgroup of H = Sp2n(K). Namely, if V = V2n(K) is the natural module for H, then we can take G to stabilize a pair E, F of maximal totally singular subspaces of V, and t to be an involution in H interchanging E and F.

Let v € Gt be a unipotent element, and let u = v2 € G. If we have E [ u = ^2 Jri (a sum of Jordan blocks) with ^ Vi = n, then since t interchanges E and F we have V j v = J2 J2ri. By [11, Lemma 6.2], V j v is a sum of indecomposables of the form W(2vi) (a sum of two singular blocks of size 2vi) and V(2vi) (a single non-degenerate block of size 2vi, appearing with multiplicity at most 2). Moreover, V j u = Y1 W(vi) because G fixes the pair E, F of singular subspaces. From Lemma 4.1 it follows that no summands V(2vi) with vi even can be present in V j v; so we may write

V j v = Y; W(2vi)ai +Y, V(2si)ci

Si odd

(an orthogonal decomposition), where each Ci ^ 2. Rewrite this as

V j v =53 W(2mi)ai + Y, W(2ni)bi + ^ V(2A*), (1)

mi odd ni even ki odd

where the mi and ni are distinct, and the ki are in non-increasing order and occur with multiplicity at most 2. The corresponding element u = v2 € G acts on the n-dimensional

space E as

E j u = £ Jm2ai + £ Jn%2hi + £ Jki. (2)

mi odd ni even ki odd

In particular, even block sizes in E j u occur with even multiplicity.

It follows from [16, pp. 21-24] (see Lemma 4.2) that any two unipotent elements v G Gt which are ff-conjugate are also G-conjugate. Hence the above decomposition V j v in (1) determines the G-class of v uniquely. Moreover, the right-hand side of (2) is also the Jordan decomposition of a unipotent element u0 of the orthogonal group On(C), and [16, p. 22] gives

dim Cg(v) = dim Con(c)(uo) + ^ 2ai, (3)

e(mi )=0

where e(mi) = 0 if there is no kj equal to mj, and 1 otherwise. The formula for dim COn(C)(u0) can be found in [11, Proposition 3.7]. Namely, rewriting the right-hand side of (2) as 0i Jiri, we have

dim COn(C) (u0) = 1 Y, iri + 53 irirj - 2 53 ri. (4)

i i<j i odd

Here is our main result about outer unipotent classes in Gt. Parts (i) and (ii) summarize the above discussion; part (iii) determines the reductive parts of centralizers, and is our new contribution.

Theorem 1.2. Assume (G,p) = (SLn(^), 2).

(i) Each decomposition (1) represents a unique G-class of unipotent elements in Gt.

(ii) For v as in (1), the dimension of CG(v) is given by (3) and (4).

(iii) The reductive part of C = CG(v) is

C/Ru(C)= n Sp2ai(K) X n I2bi(K) X Z2t+S,

mi odd ni even

hb(K) = {Sp2bi (K) if 3 kj = ni ± 1, 1 O2bi (K) otherwise,

t is the number of j such that kj > kj+i + 2 > 3, and 5 is 1 if there exists kj ^ 3 and is 0 otherwise.

We give some tables illustrating this result for n ^ 8 in Section 6.

Theorem 1.2 yields results for the finite groups GY(t) = SLn(q).2 just as in [11, Theorem 7.3], where SLn(q) = SLn(q) or SUn(q) according as e = 1 or -1.

Theorem 1.3. Assume (G,p) = (SLn(K), 2) and let y be a Frobenius morphism of G with Gy = SLn(q). Let v G Gt be unipotent and write C = CG(v). Then vG n GYt splits into 2s+t+s Gy-classes, where t and 5 are as in Theorem 1.2 and s is the number of O2bi (K) factors in C/RU(C). For x in such a GY-class, CG (x) is an extension of DY by R, where D = Ru(CG(u)) (so \Dy| = qdim D) and

R = n Sp2ai (q) X n I2bi (q) X Z2t+S,

mi odd ni even

I2b (q) = 1^2^ (q) if 3 kj = ni ± 1

1 O2b - (q) otherwise. Moreover, all 2s possibilities for R occur equally often among the 2S+t+s GY-classes.

Theorem 1.3 will be deduced from Theorem 1.2 in Subsection 4.5. 1.4. The E6 and D4 cases

Assume (G,p) = (Eg, 2) or (D4, 3). Our main results here are as follows.

Theorem 1.4. Assume (G,p) = (E6,2). Then the coset Gt contains 17 conjugacy classes of unipotent elements under the action of G, with representatives v\,... ,v\7 and centralizers given in Table 7. The conjugacy classes and centralizer orders of outer 2-elements in the coset Gy t are given in Table 9.

Theorem 1.5. Assume (G,p) = (D4, 3). Then the coset Gt contains five conjugacy classes of unipotent elements under the action of G, with representatives v\,...,v5 and centralizers given in Table 8. The conjugacy classes and centralizer orders of outer 3-elements in the coset Gyt are given in Table 10.

The tables referred to in Theorems 1.4 and 1.5 can be found in Section 6. Note that in the third column of Table 7 or 8 we list the E7-class or F4-class of each representative vi.

Layout. The rest of this paper is divided into five further sections. In Section 2, we prove Theorem 1.1. In Section 3, we establish some preliminary results, including the result of Fulman-Guralnick mentioned above. In Section 4, we prove Theorems 1.2 and 1.3, and in Section 5 we prove Theorems 1.4 and 1.5. Finally, Section 6 contains the tables for E6 and D4 referred to in Theorems 1.4 and 1.5, and also tables illustrating Theorem 1.2 for G = SLn(K) with n ^ 8. At the end there is an appendix giving extra detailed information used in the proof of Theorems 1.4 and 1.5.

2. Proof of Theorem 1.1

In this section, we assume (G,p) is any of the possibilities listed at the start of Subsection 1.1, and that t and 7 are as stated therein; we shall prove Theorem 1.1. The proof we shall give is an elaboration of Steinberg's original proof of the number of unipotent elements of a simple group of Lie type, which was based on the values of the Steinberg character. Here, we use information established in [5, 3.18] on values of an extension of the Steinberg character.

First assume that G is simply connected and p = 2, so that GY is SLn(q), ^n(q) or Eg(q), with q even. Abusing notation slightly we identify t with its restriction to GY, and form the semidirect product GY := GY (t). Define an element x € Gt to be quasi-semisimple if x normalizes a Borel subgroup of G and a maximal torus therein. By [5, 3.18], the Steinberg character of GY can be extended to GY, and the extension x satisfies the following condition for elements x € GYt :

±\Cqy (x)\2 if x is quasi-semisimple, 0 otherwise.

In the following result, by a finite group of Lie type we mean a quotient by a central subgroup of the fixed point group under a Frobenius morphism of a simple algebraic group of simply connected type. For a group X, denote by X(2) the set of 2-elements in X.

Lemma 2.1. Let A x B be the direct product of two isomorphic finite groups of Lie type A, B in characteristic 2, and let t be an involutory automorphism of A X B interchanging the two factors. Form the semidirect product (A X B)(t). Then the number of unipotent elements in the coset (A X B)t is equal to \AB\/\Cab(t)\ • \A(2)\.

Proof. Identify A and B, and write t = t(a,ß), where t is the map (a, a') ^ (a',a) (for a, a' G A) and a, ß G Aut(A). As t is an involution, we have ß = a-1.

Suppose (x,y)T is unipotent, where x,y G A. Then ((x,y)T)2 = (xya,yxß) is unipotent, so xya, yxß are unipotent elements of A. Conversely, given any y G A and any unipotent element u G A, let x = u(ya)-1; then xya = u is unipotent, as is yxß = yuß((ya)-1)ß = yußy-1. Hence the number of unipotent elements of the form (x, y)T is |A| • |A(2)| = \AB\/\Cab (t )| • |A(2)|, as required. □

Lemma 2.2. Let GY be as above, and let s G GY be a semisimple element such that Cg (s) contains a unipotent element in the coset GYt. Then Cg (s) = DR(t), where D is a commuting product of groups of Lie type in characteristic 2, R is a maximal torus of GY normalizing D, and t is a quasi-semisimple involution normalizing both R and a Borel subgroup of DR.

Proof. As G is simply connected, CG(s) is connected, so that Cq(s) = DR, where D is a product of simple algebraic groups and R is a maximal torus. We can take y to normalize R and a Borel subgroup B of D. Then Cqy< (s) = DR, where D = O2 (DY) and R = RY (see [14, 2.12]). Note that D is a commuting product of groups of Lie type. We are assuming that Cg (s) contains a unipotent element of GYt. Therefore, Cg, (s) = DR(x), where x2 G DR. Then NC _ rs)(BY) fl NC _ (s)(R) = R(t) where t is as in the statement of the lemma. Also CÖy/ (s) = DR{x) = DR{t). □

Lemma 2.3. Let s, t be as in Lemma 2.2, and write C = CGi(s) = DR. Then

\Cc(t)| = \CD(t)| • \CR/RnD(t)|.

Proof. As C = DR, we have R/R f D = C/D by an isomorphism commuting with t. Therefore, |CC/D(t)| = \CR/RnD(t)|. Also R is abelian of odd order, so that R = CR(t) x [R, t] and R f D = CRnD(t) x [R f D,t]. It follows that CR/RnD(t) = CR(t)/CRnD(t). In particular, Cc(t) covers Cc/d(t). Hence \Cr/rcid(t)| = |Cc/D(t)| = CC(t)\/\CD(t)|, as required. □

Write Z = Z(Gy). In the next two lemmas we count the elements of GY in two different ways.

Lemma 2.4. We have \GY\ = E1 + S2 + S3, where

S1 = ^^ \Cq (s)(2)|, sum over s G GY \ Z semisimple,

£2 = |Z| • \G7(2)|, S3 = \Gy(2)| - \Gy(2)|.

Proof. Observe that Si is the number of elements in GY with semisimple part s / Z, S2 is the number of elements in GY with semisimple part in Z, and is the number of remaining

elements in GY (since CZ (t) = 1).

Lemma 2.5. We have \GY\ = Aa + Ab + AC + Ad, where

(s)\2) , sum over s e G7 \ Z .semisimple,

Ab = (s,x)\2)2, sum over sx e GY quasi-semisimple with

semisimple part s e GY \ Z, Ac = \Z\(\Gy\2)2,

Ad = (x)\2)2, sum over x e GY quasi-semisimple of order 2.

Proof. Let x be the Steinberg character of GY. Then (x,x) = 1 implies that \GY\ = Thgeo x(g)2, and we have simply broken up the sum into the four parts Aa,..., Ad, where Aa and Ac arise from applying the usual Steinberg character to elements of GY, while Ab and Ad arise from applying the extended character to elements in GYt . □

From the previous two lemmas, we have

Si + + £3 = Aa + + AC + Ad ■

Lemma 2.6. We have Si = Aa + Ab.

Proof. Obviously, Si = + Si// where

Si' = £ |CG^(S)(2)|, Si// = £ |CÖ7(S)(2)| - |CG^(S)(2)|.

By Steinberg's formula [4, Theorem 6.6.1] for the number of unipotent elements in each group CGj (s), we have Si/ = Aa; so we need to show that Si// = Ab.

Let s e Gy \ Z be a fixed semisimple element and write C = CGy (s). As in Lemma 2.2 we have Cg (s) = C(t) = DR(t). An outer unipotent element in C(t) projects to an involution in C(t)/D, and these have the form Drt with rt e Rt an involution. Such involutions rt are all R-conjugate to t, so the total number of outer unipotent elements in the coset Ct is the number in Dt times the number of cosets Drt.

Now t acts on the set of Lie-type factors of D, with each factor being either normalized or interchanged with an isomorphic factor. Let D be the universal cover of D, so that D is a direct product of Lie-type factors and we can pull back the action of t. Then using induction together with Lemma 2.1, we see that the number of unipotent elements in the coset Dt is equal to

Cd (t)|

(|C b (t)|2 )2.

Write D = D/J, where J is abelian of odd order, and let J- be the set of elements in J inverted by t. Then each unipotent element in Dt pulls back to J-| unipotent elements in Dt, all of which are conjugate under J-. As (|C^(t)|2)2 = (|CD(t)|2)2, we see that the number of

unipotent elements in the coset Dt is equal to

Mi (t)b>2'

The number of cosets Drt with rt an involution is \R/R n D\/\CR/RnD(t)| = | C/D\/\CR/RnD(t) \ . Hence by Lemma 2.3 and the above equation, the number of unipotent elements in the coset Ct equals (\C\/\CC(t)\)(\CD(t)\2)2 = (\C\/\CC(i)^^^^(s,t)\2)2. The quasi-semisimple elements sx G GY having semisimple part s and x of order 2 are all conjugate to st. So, summing over s we obtain Si" = Ab, as required. □

We now complete the proof of Theorem 1.1 for the cases where G is simply connected and p = 2. By Steinberg's formula, we have S2 = Ac, and hence by (5) and Lemma 2.6, S3 = Ad. All the quasi-semisimple involutions x G GY are conjugate to t, so Ad = \GY : GY,T\ ■ (\GYtT\2)2. Since S3 is the number of unipotent elements in GYt, the result follows.

Next we relax the assumption that G is simply connected. Let G be the simply connected cover of G. We can consider 7 and t acting on G. There is a natural surjection GY ^ Op (GY). Let X = Gy/Z(Gy) = Op'(Gy)/Z(Op (GY)). Using [15, 2.3 and 2.4], we see that \Z(GY)\ = \Z(Gy)\ ■ (\Gy\/\Op (Gy)\). Moreover, the outer unipotent elements in GYt invert both Z(GY) and Gy/Op (Gy). The above argument with D and D shows that both the number of outer unipotent elements and the formula in Theorem 1.1 are independent of the form of G. Indeed, in either case the number of outer unipotent elements equals \Z(GY)\ times the number of outer unipotent elements in Xt. We note that the theorem also holds if we replace GY by Op (Gy).

It remains to handle the p = 3 case of Theorem 1.1. Starting with G = D4 simply connected this can be achieved by arguing as above with obvious changes along the way, and this is left to the reader. (We note that the only place requiring an analogue of Lemma 2.1 is where t transitively permutes the factors in the direct product of three copies of SL2; so D = D in this situation.) The argument of the above paragraph then gives the result when G is an adjoint group. Alternatively, the result can be established easily from the information given in Table 10 in Section 6.

3. Preliminary lemmas

In this section, we assume (G,p) = (Ai, 2), (E6, 2) or (D4, 3). We shall first prove a special case of a result of Fulman and Guralnick [6, Lemma 2.2], and then establish some basic lemmas which combine this result with information from [11]. In particular, we count the number of Gy-classes of p-elements in GYt. For each p-element in Gt we then introduce a parabolic subgroup of G which will play an important role in the determination of the G-centralizer.

Lemma 3.1. Let X be a finite group with normal subgroup Y of index p, and let t g X \ Y have order p. Then the number of X-orbits on p-elements in Yt equals the number of t-stable Y-orbits on p-elements in Y.

Proof. Observe first that if x G X, then

xY is T-stable ^^ xY = xX ^^ CX (x) n Yt = 0.

P1 = {p-elements in Yt}, P2 = {p-elements in Y with T-stable Y-orbit}.

Then for i = 1, 2 the X-orbits on Pi are the same as the Y-orbits on P.i; write n for this common number of orbits. We seek to show that n1 = n2.

whence

Take i e {1, 2}; given x e X write Fi(x) = \Cx(x) n Pi\. For x e X \ Y, the generators of (x) are distributed in equal numbers among the cosets YTj for j = 0, so we have both

1 1 p-i 1

n = m E Fi(x) = m E Y.Fi(vTj) = r^E(Fi(y) + (P - 1)Fi(yT))

1 1 xex 1 1 yeY j=0 1 1 yeY

ni = Y E Fi(y),

1 1 yer

ni = ME Fi(yT).

1 1 yer

Now take x = yT e Yt; choose k = 1 (mod p) such that z = xk is a p-element, so certainly 2 e Cx (x) n Yt. Thus, the map g ^ zg is a bijection CY (x) ^ Cx (x) n Yt; and g e CY(x) is a p-element if and only if zg e Cx (x) n Yt is a p-element. Since any element of CY (x) has a t-stable Y-orbit by the initial observation, we thus have a bijection Cx (x) n P2 ^ Cx (x) n P1, whence F2(x) = F1(x). It follows that n2 = n1 as required. □

Our next two results show (as is done in [6, 2.6]) that the finiteness of the number of outer unipotent classes in the coset Gt follows from combining Lemma 3.1 with the finiteness of the number of unipotent classes in G.

Let p(n) denote the number of partitions of n.

Lemma 3.2. Assume G is adjoint.

(i) If (G,p) = (Ai, 2), then GY has p(l + 1) conjugacy classes of 2-elements, each of which is t-stable.

(ii) If (G,p) = (E6, 2), then GY has 28 conjugacy classes of 2-elements, each of which is t-stable.

(iii) If (G,p) = (D4, 3), then GY has 7 conjugacy classes of 3-elements which are t-stable.

Proof. (i) We know that G has p(l + 1) classes of unipotent elements and [11, Theorem 7.1] shows that the same holds for the finite linear and unitary groups.

(ii) The first assertion follows from [11, Corollary 17.7]. For the second assertion, we make use of the explicit information on centralizers of unipotent elements of G(q) presented in [11, Table 22.2.3]. Let u be a unipotent element of GY, and write u' = uT. We claim that u' = ug for some element g e GY.

Set Cu = CGy (u) and Cu> = CGj (u'). By comparing first the sizes of O2(Cu) and O2(Cu>), and then the orders of Cu and Cu/, we see that u and u' are GY-conjugate except possibly for the cases where u has G-class D4, D5, E6 or Ee(a3). In each of these cases the G-class splits into two Gy-classes of equal size; so the only question is whether t fixes each of the two classes or interchanges them. Now GY,T = F4(q) contains unipotent elements of type B3, B4, F4, and these elements lie in the G-classes D4, D5, E6, respectively; so, t must fix each of these G-classes. Also if u has type F4(a2) in F4, then we see from [11, Table 22.1.4] that u lies in the G-class E6(a3); moreover, a representative for the class of u can be taken over the prime field (see the discussion at the start of [11, Chapter 18]). So, here too t fixes each of the classes, establishing the claim. This completes the proof of (ii).

(iii) Here we begin at the level of the orthogonal group SO8 (K), whose unipotent classes are tabulated in [11, Table 8.5a]. Those with distinguished normal form W(1)4, W(2) + W(1)2,

W(3) + W(1), W(2) + V(3) + V(1), V(5) + V(3), V(7) + V(1) have Bala-Carter labels 1, A1, A2, D2A1 = A13, D4(a1), D4, respectively, while those with distinguished normal form W(2)2 (two classes), W(1)2 + V(3) + V(1), W(4) (two classes), W(1) + V(5) + V(1) have Bala-Carter labels (A12)' or (A12)", D2, (A3)' or (A3)", D3, respectively. Now take the corresponding classes in G. In each of the first six cases the triality automorphism t stabilizes a root subsystem of the relevant type, and hence the class of regular (or in one case subregular) unipotent elements in the corresponding subsystem subgroup; in the second six cases, however, t cycles the first three and the second three classes. By [11, Theorem 3.1], the centralizer for the A2 class has component group of order 2, while that for each of the other five T-stable classes is connected; descending to the finite group GY we see that (iii) holds. □

Lemma 3.3.

(i) If G is adjoint, then according as (G,p) = (Al, 2), (E6, 2) or (D4, 3) there are p(l + 1), 28 or 7 Gy-classes of p-elements in GYt .

(ii) There are finitely many G-classes of p-elements in Gt (for G of arbitrary isogeny type, provided G admits t in the case (G,p) = (D4, 3)).

Proof. Part (i) follows from Lemmas 3.1 and 3.2. For (ii), we first observe that it suffices to assume G is adjoint by the final paragraph of Subsection 1.1; moreover, we can apply [8, Proposition 1.1] to assume that K is the algebraic closure of the prime field. We claim that the number of G-classes of p-elements in Gt is at most k, where k = p(l + 1), 28 or 7, respectively (later we shall see that there are precisely 17 such classes if G = E6 and 5 classes if G = D4). If this claim were false, there would be at least k + 1 such classes, and we could choose a sufficiently large power q of p such that each class had a representative in GYt, where 7 = a for a the q-field morphism as in Subsection 1.1; but this would contradict (i). Therefore, the claim holds and (ii) is established. □

Let v be a p-element in Gt, and write u = vp G G. We bring into play a parabolic subgroup of G which contains CG(u) and for which a certain density statement holds. In order to do this we make use of a certain nilpotent element e in the Lie algebra L(G). For G = Al we can view G as SLl+1(K) and let e = u — 1. For G = E6, we use representatives of the unipotent classes of G given in [11]; we can take u to be a product of root elements u = J} upi for certain 3. G S, and then u and the corresponding nilpotent element e = ^ ept are linked (see [11, p. 281 and Theorem 17.3]). For G = D4, this is described in [11, Lemmas 2.15, 3.13].

As is explained in [11, p. 4], there is a certain 1-dimensional torus T in G which acts by weight 2 on (e); by taking an appropriate simple system we may assume for each simple root 3 the T-weight on (ep) is 0, 1 or 2, and we write A for the corresponding labelled Dynkin diagram. The torus T determines a parabolic subgroup P = QL, where L = CG(T) and Q = n Xp with the product taken over those roots 3 G S for which the T-weight on (ep) is positive; thus the simple roots of L are those with label 0 in A. We write Q^2 (respectively Q>2) for the subgroup Xp of Q, where the product is taken over those roots 3 such that the T-weight on (ep) is at least 2 (respectively, greater than 2). Then for each root element up appearing in the expression for u, the T-weight on (ep) is 2, so that u G Q^2 (see [11, (18.1)]).

If (G,p) = (E6, 2), the correspondence between unipotent classes uG and labelled Dynkin diagrams A is given in [11, Table 22.1.3]. If instead (G,p) = (Al, 2) or (D4, 3), the labelled Dynkin diagram A can be obtained from the action of T on the natural module W = VG(A1) as follows. We may assume T is contained in the maximal torus TG of G; then the TG-weights on W determine the T-weights on the simple roots a.. In the Al case the TG-weights on W are A1, A1 — a1, A1 — a1 — a2,...; thus, if for example l = 9 and u (and hence e) acts as

J5 + J3 + J2, the T-weights on the individual blocks are (4, 2,0, -2, -4), (2, 0, -2), (1, -1), so that the T-weights on W are 4, 22,1, 02, -1, -22, -4, and hence A is 201101102. In the D4 case the TG-weights on W are Ai,Ai - ai,Ai - ai - a2,Ai - ai - a2 - a3, Ai - ai - a2 - a4,...; thus, if for example u (and hence e) has distinguished normal form W(2) + V(3) + V(1) and so acts as J3 + J22 + Ji, the T-weights on W are 2,12, 02, -12, -2, and hence A is ioi.

The following lemma follows from [11, Theorem 1] for G = Al, D4 and [11, Theorems 17.4, 17.5] for G = E6.

Lemma 3.4.

(iii) uQ>2 is fused under the action of Q.

Our final result in this section shows that if the parabolic subgroup P is T-stable then it is stabilized by v.

Lemma 3.5. Assume that P = PT. With v, u and P as above, we have P = Pv.

Proof. We have u e Q^2 < P, and uP is dense in Q^2. By hypothesis t normalizes P; thus Pv = Pg for some elementg e G, and so Qv = Qg and (Q')v = (Q')g. It follows from [2] that Q^2 = Q or Q', since the labels in the labelled Dynkin diagram A which determines P are 0,1, 2. In either case u e (Q^2)v = (Q^2)g and this group is either Qg or (Q')g.

Now dim CP(u) = dim CPv (uv) = dimCPv (u) = dim CPg (u). A dimension comparison implies that uP is dense in (Q^2)g. But (ug)P is also dense in (Q^2)g. Therefore, (ug)pS = u for some element p e P, so that pg e CG (u) = CP(u). Hence, g e P and P = Pg = Pv as required. □

In this section, we assume (G,p) = (Al, 2). Most of this section is devoted to the proof of Theorem 1.2. Finally Theorem 1.3 is deduced in Subsection 4.5.

4.1. Preliminaries

As in Subsection 1.3 write n = l + 1 and take G to be simply connected, so that G = SLn(K); regard G(t) as a subgroup of H = Sp2n(K). Let V = V2n(K) be the natural module for H with symplectic form ( , ), and write V = E © ET, where G acts on each of the totally singular summands E and ET. First note that the fact that any unipotent element u G acts on E and ET forces V j u to be a sum of summands of type W(k). The following result justifies Equation (1) of Subsection 1.3.

Lemma 4.1. Let v e Sp(V) be a unipotent element such that V j v = V(2k). Then

4. Proof of Theorems 1.2 and 1.3

V j v2

V(k)2 if k is even, W(k) if k is odd.

Proof. Here v acts on V as follows. Take a basis v2k,v2k-i,...,vi of V(2k), where the symplectic bilinear form is given by (vi, vj) = 1 if i + j = 2k + 1 and 0 otherwise. Now define

v to act as

Vk+i --► Vk+i + Vk+i-1 +-----+ Vk (1 < i < k),

Vj -—> Vj + Vj-1 (2 ^ j ^ k), vi -—> V1.

Clearly v2 has two Jordan blocks of size k. The question is whether these blocks are singular or non-degenerate (meaning that v2 acts on V as W(k) or V(k)2, respectively).

Assume first that k is even. For x G V write [x, v2] = x + xv2, [x,v2](2) = [[x,v2],v2], and in general [x, v2](i) = [••• [[x, v2], v2],..., v2], the i-fold commutator. We compute inductively that for i ^ k/2 — 1, the commutator [v2k, v2](i) is of the form:

V2k-2i + Vj Vj + 53 Vl Vl.

j even, 2k-2i>j^k l<k

In particular,

[V2k,v2](k/2-1) = Vk+2 + VkVk + Vivi.

The next commutator is then of the form

[V2k ,V2](k/2) = Vk + Vk-1 + ^ VlVl.

As [vi, v2] = vi-2 when 3 ^ i ^ k, it follows that

[V2k ,V2](k-1) = V2 + V1.

Since (v2k,v2 + v1) = 0, it follows that the Jordan block for v2 generated by v2k is non-degenerate, hence is of type V(k). Also V(k)v is another Jordan block for v2 of the same type and orthogonal to V(k); so V I v2 = V(k)2 when k is even.

Now assume k is odd. In this case, we compute inductively that for any i, the commutator [v2i, V2] involves only terms in Vj for j ^ 2i — 2 even. Hence [v2k,v2](k-1) = v2, and so as (v2k,v2) = 0, the Jordan block for v2 generated by v2k is singular of size k. Applying v to this block gives another singular block, and hence V j v2 = W(k) when k is odd. For use in the next lemma we note that [v2k, ([v2k,w](k-1))v] = [v2k,V2v] = [v2k,V2 + V1] = 1 = 0. □

In [16, p. 21], Spaltenstein views elements of Gt as non-degenerate bilinear forms on E and goes on to describe the conjugacy classes with respect to these forms. We can also view elements of Gt in this way as follows. Let v and u be as in equations (1) and (2) of Subsection 1.3. Define a non-degenerate bilinear form ( , )v on E by (e1,e2)v = (e1 ,e2v). Let A be the partition of n determined from the action of u on E and define a function e : N ^{0,1, u} which for all i sends

ni -—► u, ki 1,

{1 if 3 kj = mi, 0 otherwise,

and sends all other positive integers to 0. Thus, V j v determines a pair (A, e).

Lemma 4.2. Let G(t) = SLn(K)(t) < Sp(V) = H as above. Then any two unipotent elements v G Gt which are H-conjugate are also G-conjugate.

Proof. According to [16, p. 21], the G-classes of unipotent elements in Gt are in bijective correspondence with the set of pairs (A, e), where A is a partition of n such that all even parts have even multiplicity, and e : N ^{0,1,w} is a function having various properties which are described in [16, p. 22].

We wish to show that the function defined above agrees with the one given by Spaltenstein. The key facts which we must establish are that if w is in a W(2m) summand of V j v, then (w, ([w, u](m-1))v) = 0, whereas in a V(2m) summand with m odd there exists w for which this does not hold. The latter was established at the end of the proof of Lemma 4.1.

Consider a summand of type W(2m). The action of v is given in [11, p. 92]. With a slight change of notation, we see that v has two Jordan blocks of size 2m with bases xi,..., x2m and yi,..., y2m, where xi and yi generate the blocks and x2m and y2m are fixed points. One shows by induction that for i < 2m - 1 odd, we have [xi, u] = xi+2 + wi, where wi is a sum of terms xj for j > i + 2 odd. It follows that the Jordan block for u generated by xi has fixed space spanned by x2m-i = [xi, u](m-1). A similar conclusion holds for the block generated by yi. Now u has four Jordan blocks of size m on W(2m), namely the two just described and their images under v. It is now straightforward to check that (w, ([w,u](m-1))v) = 0 for all w e W(2m).

At this point, we see that the function e defined above agrees with the one defined in [16, p. 22]. Therefore, the restriction V j v uniquely determines e. The result follows. □

4.2. Some special cases

The next few lemmas prove Theorem 1.2 in some key special cases. The first lemma settles two base cases which will be needed in a later inductive proof.

Lemma 4.3. If V j v is either W(4) + V(2) or W(4) + V(6), then C°/Ru(C) = Sp2.

Proof. By [11, Theorem 4.2] the full centralizer of v in Sp(V) has its connected reductive part Sp2. Therefore, C°/Ru(C) is contained in Sp2. Suppose that equality holds for the case V j v = W(4) + V(6). Let W = [V, v](4). Then C acts on W±/W and v acts on this space as W(4) + V(2). So the equality for W(4) + V(2) also holds.

Now consider W(4) + V(6), where G = SL7(K) = A6 with fundamental roots ai,..., a6. We use notation as in Subsection 1.1 and set v = tu111000u011100. One checks that u = v2 lies in the class A2A12, and so has Jordan form J3 + J22. Therefore v is necessarily in the correct class.

The labelled Dynkin diagram A corresponding to u is 101101; thus, the parabolic subgroup P = QL described prior to Lemma 3.4 has Levi factor of type A1A1. Using notation xijk (d) = xai+aj +ak (d) and so forth one checks either by hand or by the computer technique described in the appendix that

Cu (v) = {x2(ti2 )x5(ti2)x34(t2)xi2(ti)x56(ti)x23(ti)x45 (ti) x X2345(t3)xi23(C)X456(C^23456^4)^234(¿5^345 (¿5) X Xi234(¿6)x3456(¿6)xi2345(tr)x23456(¿i + ¿7) : Z € F2,ti € K}.

Then CU(v)Q/Q = {x2(f)x5(f) : f € K}Q/Q, a 1-dimensional unipotent subgroup of P/Q. In

addition s2s5xoiiioo(1) € CP(¿), which implies that CP(¿)Q/Q contains SL2 = Sp2, as required.

Lemma 4.4. Let v e SLn(T) < Sp(V) be such that V j v = W(2m)a for some m, and let C = CSLn (v). Then

C/Ru(C)={Sp2° if m isodd, O2a if m is even.

Proof. We can regard V j v as V(2m) ® I2a ^ Sp2m ® Sp2a, acting on a pair of singular spaces of the form V(2m) ® R and V(2m) ® S. In Sp(V), the element v is obviously centralized by the factor Sp2a, and this is the full reductive part of the centralizer, by [11, Lemma 6.7].

If m is odd, we easily obtain the result as follows. For then Lemma 4.1 implies that u = v2 acts on V as (Jm + Jm') ® I2a = W(m) ® I2a, with v interchanging the two singular blocks Jm and Jm'. Replace the above SLn by the Sp(V)-conjugate which stabilizes each of the singular spaces Jm ® I2a, Jm' ® I2a. Then v normalizes this SLn and since the Sp(V)-class determines the SLn-class, v has the same centralizer in each SLn. As v clearly centralizes an Sp2a subgroup in the second SLn, this gives the assertion when m is odd.

Now suppose m is even. Here, Lemma 4.1 shows that u = v2 acts on V as (Jm + Jm') ® I2a = (V(m) + V(m)') ® I2a, where this time the blocks V(m) and V(m)' are non-degenerate and interchanged by v. Let V2a be the natural module for the factor Sp2a, and choose singular a-spaces R, S in V2a such that V2a = R + S. Define

W = (V(m) ® R) + (V(m)' ® S), W' = (V(m) ® S) + (V(m)' ® R).

These are singular 2am-spaces in V interchanged by v. We take our group SLn(T) = SLn(v) in the stabilizer in Sp(V) of the pair {W,W'}. The centralizer CSLn (v) clearly contains the subgroup GLa.2 of Sp2a which fixes or interchanges the pair R, S. Hence, GLa.2 ^ C/Ru(C) ^

Next we claim that when a = 2 we have C/Ru (C) ^ O4. So, suppose a = 2 and V j v = W(2m)2. Then u = v2 is in the class of Jm4 in G = SLn. We now work with root groups in the root system of G of type A4m-1, with fundamental roots a1,..., a4m-1. Take t to be a standard graph automorphism of G. Write uijk... = xai+aj+ak+...(1), and define

v' = T(ui234u5678 ' ' •u4m-7,4m-6,4m-5,4m-4) X («2345^6789 • • • «4m-6,4m-5,4m-4,4m-3).

(v')2 = (m1234«5678 • • • U4m-7,4m-6,4m-5,4m-4)

x (u2345u6789 • • • u4m-6,4m-5,4m-4,4m-3) x (u4m-4,4m-3,4m-2,4m-i • • • «4567) x (u4m-5,4m-4,4m-3,4m-2 • • • «3456),

which is in the class (Am-1)4 in SLn, hence is conjugate to u. Since W(2m)2 is the only

Sp(V)-class squaring to u, we may therefore take v the group

v'. One checks that v is centralized by

(xl(t)xз(t) • • •X4m-i(i), S1S3S5 • • • sm-i : i€K) = A = Ai,

and also by the element

j = S2S6«10 • • • «4m-2u2345u6789 • • • u4m_6,4m-5,4m-4,4m-3.

Moreover, A commutes with Aj, so CG(v) contains (A, Aj) = SO4. Since we already know that C/Ru(C) contains GL2.2, it follows that C/Ru(C) contains O4, as claimed.

Now we return to the case where a is arbitrary, and argue that C/Ru(C) contains O2a. To see this, we can suppose that a ^ 2 (since if a =1 then C contains GL1.2 = O2). Write

V j v = W(2m)2 + W(2m)a-2,

and let E1, F1 be the corresponding pair of singular 4m-spaces for the first factor, and E2, F2 those for the second. So, we can take G = SL2am to fix the pair Y = E1 + E2, Z = F1 + F2 of singular spaces in V. By the above claim, C/Ru(C) contains O4 acting on E1 and F1, and centralizing E2 + F2. It also contains GLa.2 and is contained in Sp2a. As the only proper overgroups of GLa.2 in Sp2a are O2a and Sp2a, it follows that C/Ru(C) contains O2a.

Finally, we argue that C/Ru(C) = O2a. Suppose this is not the case. Then C/Ru(C) = Sp2a. Let T be a torus of rank a - 1 in C centralizing a subgroup Sp2 of C/Ru(C), and let X be a minimal preimage of this Sp2 in CC (T) (that is, As above, let Y, Z be a pair

of singular spaces in V fixed by G = SL2am and interchanged by v. Then Y = CY(T) + [Y, T], and X acts on both these subspaces. In [Y, T] there are 2a - 2 weight spaces of T, each of dimension m, and u = v2 acts on each as a single Jordan block. As X acts on each weight space, centralizing the action of u, it follows that X induces a unipotent group on each weight space, and hence that X acts trivially on [Y, T]. Now consider CY (T). The fixed point space of T on V is CY(T) + CY(T)v, a non-degenerate 4m-space on which v acts as W(2m). Moreover, Cg(v) induces Sp2 modulo its unipotent radical on this space. Hence, we have reduced to the case where a = 1.

So assume that a =1. Next, we further reduce to the case where m = 2. Regard V j v as V(2m) ® V2, with v trivial on V2. Here, v G Sp(V(2m)) acts on the first factor and is centralized by Sp(V2) acting on the second factor. First note that, given our supposition, C induces Sp2 on each C-composition factor of V. This is because C covers Sp(V2) modulo Ru(CSp(V)(v)) and Sp(V2) acts homogeneously on V.

Let R, S be singular 1-spaces in V2 as before. There is an involution t G Sp(V2) interchanging R and S. Then V = vt interchanges the two singular spaces W = V(2m) ® R and W' = V(2m) ® S. Also t centralizes Sp(V2m) and therefore t induces a standard graph automorphism of G = GL(W). Also (vt)2 = v2t2 = u, which induces Jm2 on W. Hence we may replace v by V = vt, and take G = GL(W) with t = t. Then our supposition implies that C = CG(V) has reductive part Sp2 which acts on each C-composition factor as a natural module.

Let F = CV(2m)(v), a 1-space. Then CV(v) = F ® V2. As t commutes with v, we see that t acts on CV(v)^/Cy(v) = (F^ ® V2)/(F ® V2). As V2 = u, the element which V induces on V(2m - 2) ® V2 is in the class W(2m - 2), and still has Sp2 in the reductive part of its centralizer in GL(V(2m - 2) ® R). Repeating this argument, we end up with an element in the class W(4) in Sp8, lying in GL4.2, and such that its centralizer in GL4 has reductive part Sp2. Thus, we have now reduced to the case where m = 2.

Now observe that SL4.2 = O6, and v is in the class V(4) + W(1) of O6. Indeed this follows from [11, (4.4) and Theorem 4.2], since v is an outer unipotent element of order 4 centralizing a T1 < SL4. But then [11, Theorem 4.2] also shows that Cso6(v)/Ru(Cso6(v)) = O2 rather than Sp2 for this class. This final contradiction completes the proof. □

Lemma 4.5. Let v g SLn{t) < Sp(V) be such that V j v = W(2m)a + V(2m ± 2) with m even, and let C = CSLn(v). Then C°/Ru(C) = Sp2a.

Proof. We know by Lemma 4.4 that C/Ru(C) contains O2a, and it is contained in Sp2a since this is the reductive part of the centralizer of v in Sp(V) (see [11, Theorem 4.2]). Hence, it is enough to show that C/Ru(C) contains a long root group of Sp2a, and for this it is sufficient to consider the case where a = 1. We do this by induction on the dimension of V, the base cases W(4) + V(2) and W(4) + V(6) being handled by Lemma 4.3.

Assume first that V j v = W(2m) + V(2m + 2) with m> 2. We take G{t) = SL(R){v), where R, Rv are maximal singular spaces in V interchanged by v, and V = R + Rv. Then dimR = 3m + 1, and u = v2 acts on R as Jm+1 + Jm2. We work with root groups in the root system of G of type A3m, with fundamental roots a1,..., a3m. Taking t to be a standard graph automorphism, and writing uijk... = xai+aj +afc+...(1), define

v' = T(u123u456 • • • u(3/2)m-2,(3/2)m-1,(3/2)m)(u234u567 • • • u3m-4,3m-3,3m-2). One checks that

u' = (v')2 = (u234u567 • • • u3m-4,3m-3,3m-2)

X (u345u678 • ••u3m-3,3m-2,3m-1) 1 X (u123u456 • • • u(3/2)m-2,(3/2)m-1,(3/2)m)T X (u123u456 • • •u(3/2)m-2,(3/2)m-1,(3/2)m).

Then u' = (v')2 is in the class Am(Am-1)2 of G, hence is conjugate to u. So, we may take v = v' and u = u'. Let T be the 1-dimensional torus described in the discussion preceding Lemma 3.4 which acts with weight 2 on each root element eijk corresponding to a root group element ujk appearing in the expression for u. The corresponding labelled Dynkin diagram A has label 0 on the nodes a2,a5,a8,..., a3m-1 and 1 on the other nodes. Let P = QL be the parabolic subgroup of G determined by A, such that L = CG(T) and Q is the product of all root groups for a maximal torus in L having positive T-weight. By Lemma 3.4, CG(u) < P. For c G K, define

t(c) = x-2 (c)x-5 (c)x-8 (c) • • •x_(3m-1)(c),

a root element in a diagonal A1 subgroup of L. We shall see that this element can be adjusted to centralize v. Setting

r(c) = x34(c)x67(c) • • • x3m/2-3,3m/2-2(c),

one checks that

Vt(c)r(c) = Vx3m/2,3m/2 + 1(c).

Then vt(c)r(c)s(d) = v, where s(d) = u3m/2(d)u3m/2+1(d) and d2 = c. It follows that CQ/Q contains a root group of a diagonal A1 of the Levi factor. Now CQ/Q < CG(u)Q/Q which is A1T1 by [11, Theorems 1 and 3.1]. Therefore, CQ/Q contains {O2,t(c) : c G K) = Sp2, as required.

Now consider the case where V j v = W(2m) + V(2m - 2) with m > 2. The argument is very similar to the above. Here, u = v2 acts on the singular space R as Jm-1 + Jm2. Working in the root system of G of type A3m-2, we may take

V = T(u123u456 • • • u3m—5,3m—4,3m—3 )(u345u678 • • • u(3/2)m-3,(3/2)m-2,(3/2)m-1)-

The 1-dimensional torus T has labelled Dynkin diagram A with label 0 on the nodes a1,a4,a7,...,a3m-2 and 1 on the other nodes. Write P = QL for the parabolic subgroup determined by A as above. Define

t(c) = x-1(c)x-4(c)x-7(c) • • • x-(3m-2)(c)

r(c) = x23(c)x56(c) • • • x3m/2-4,3m/2-3(c).

Vt(c)r(c) = Vx3m/2-1,3m/2(c) .

Finally, vt(c)r(c)s(d) = v, where s(d) = x3m/2-1(d)x3m/2 (d) and d2 = c. As before this yields the assertion, completing the proof. □

Lemma 4.6. Let v e SLn(t) < Sp(V) be such that V j v = W(2m)a + £ V(2ki)c* with m even, ki = m ± 1 for all i, k1 > k2 > • •• , and ci ^ 2 for each i. If C = CSLn (v), then C°/Ru(C) = SO2a.

Proof. Lemma 4.4 implies that C°/Ru(C) contains SO2a, and it is contained in Sp2a since this is the reductive part of the centralizer of v in Sp(V). Hence, C°/Ru(C) is Sp2a or SO2a. We proceed by induction assuming V is a counterexample of minimal dimension. That is, suppose that C°/Ru(C) = Sp2a.

If a > 1, let Ta-1 x T1 be a maximal torus of SO2a (acting on W(2m)2a) such that Ta-1 < SO2a-2 and Ti = SO2. Then Cc(Ta-i) acts on Cy(Ta-i) = W(2m) + £ V(2ki). Also CC (Ta-i) contains T1 and has a quotient of type Sp2. So by restricting to Cy (Ta-1) it is enough to obtain a contradiction for the case V j v = W(2m) + £ V(2ki)ci with C°/Ru(C) = Sp2.

In the course of the proof to follow we will produce certain sections A/B of V on which v and C act, and then contradict minimality. We note the reductive part of the action of C will still contain Sp2, provided the summand W(2m) is not contained in B. This follows since a maximal torus of C acts without fixed points on W(2m) (and acts trivially on all other summands).

Assume that m > 2. If J is a Jordan block for v of size greater than 2 in the above decomposition, then [J, v](2) = [J, u] has codimension 2 and Cj(u) has dimension 2. The quotient is a section of J of dimension 4 less than that of J. Now do this for all of V. Let X = [V, v](2) = [V, u] and Y = Cy u](u). Then C acts on X/Y, a non-degenerate space such that (X/Y) j v = W(2m - 4) + £ V(2ki - 4)ci, where the sum is over those i for which ki > 1. Moreover, C° contains a factor Sp2 in its action on X, which contradicts the minimality of dim V. Hence, we reduce to the case m = 2, where by hypothesis, ki = 1, 3. Therefore, we now have V j v = W(4) + £ V(2ki)c*, with ki > 5 for each i.

If k1 > 5, we can again use the inductive hypothesis to get a contradiction. Indeed, let W = [V,v](2kl-2). Then W = Cy(2kl)c1 (u) is a sum of c1 Jordan blocks J2 for v in V(2k1)ci. Now consider the action of v on W±/W. There is a complication if k1 - 2 = k2 and c1 + c2 > 2. If this does not occur, then we contradict the minimality of dim V. So consider the exceptional case where (WL/W) j v = W(4) + W(2k2) + V(2k2)ci+c— + £> V(2ki)c*. The proof of Lemma 4.4 shows that the centralizer of the induced action of v contains subgroups E = Sp2 and O2 acting faithfully on W(2k2) and W(4), respectively, and acting trivially on the remaining factors. As these groups commute, the reductive part of the centralizer of the induced action of v is Sp2 x Sp2 with C° covering one of the factors and E covering the other. So, if D is a maximal torus of E, then the centralizer of D acts on [W^/W, D] and has reductive part Sp2. Hence, we obtain a smaller counterexample on a space where v acts as W(4) + V(2k2)ci+c2-2 + £i>2 V(2ki)c*.

The final case to consider is that where V j v = W(4) + V(10)a. Here, we set S = [V, v](5), a maximal singular subspace of the summand V(10)a. Then (S±/S) j v = W(4) and C acts such that the reductive part of the induced action is Sp2, contradicting Lemma 4.4. □

4.3. Connected centralizers in Theorem 1.2

We now start work on the proof of Theorem 1.2 in general. Thus, we continue to assume G = SL„(K) with G(t) < Sp(U); let v € Gt, and as in (1) write

V i v = £ W(2mi)a + W(2m)bi + E V№),

mi odd ni even ki odd

where the mi and ni are distinct, and the ki are in non-increasing order and occur with multiplicity at most 2. Let C = CG(v) and C = C/Ru(C).

We now identify the connected reductive group C°. This contains Sp2ai X SO2bi by Lemma 4.4, and is contained in nSp2ai x Sp2bi (since this is the connected reductive part of the centralizer of v in Sp(V)). So,

C° = n Sp2ai X n I2bi,

where each I2b. is Sp2bi or SO2bi. Let T(i) be a maximal torus of the product of all factors of C° apart from I2bi, and letCi = cC(T(i)). Then C°°/Ru(Ci) = I2bi, and Ci acts on CV(T(i)), which is just the space W(2ni)bi +J2 V(2ki). Hence, by Lemmas 4.5 and 4.6, we have I2bi = Sp2bi if there exists kj = ni ± 1, and I2bi = SO2bi otherwise. Hence, CC° is as in the statement of Theorem 1.2(iii).

4.4. Component groups of centralizers in Theorem 1.2

Continuing with the notation of the previous subsection, to complete the proof of Theorem 1.2 it remains to find the component group C/C° of C = CG(v). This is given in [16, p. 24]. However, there is a small error there, so we shall offer an independent proof. Let v and u be as in Equations (1) and (2) of Subsection 1.3 and let A and e be as described in the discussion prior to Lemma 4.2.

Let us first explain the assertions and the error in [16, p. 24]. Relabel the partition of n given by (2) as (A1, A2, A3,...) with A1 ^ A2 ^ • • • , so E j u = ^ Jxi. Then the component group C/C° is the abelian group generated by a set of involutions {ai : e(Ai) = 0}, subject to the following relations: aiaj = 1 if Ai = Aj, or if Ai = Aj + 1, or if Ai is odd and Ai = Aj +2; and also ai = 1 if Ai = 1. The error in [16] is that the last relation is omitted. One checks that this description of the component group agrees with the one given by Theorem 1.2.

We now present our proof that the component group C/C° is as asserted in Theorem 1.2. As before, regard SLn.2 = SL„{t) < Sp2n, where t is the standard graph automorphism of G = SLn with respect to a fixed root system. Let V be the natural module for Sp2n and write V = E © ET, where G = SLn acts on each of the totally singular summands. Recall that we have a non-degenerate bilinear form ( , )v on E, where (e1, e2)v = (e1, e2v).

Let S < E. Then S is a singular space under the form ( , )v if and only if S + Sv is singular in V. Let S±v denote the annihilator in E of S under ( , )v.

Let v G Gt with u = v2. Write V j v and E j u as in (1) and (2), respectively. We shall see in Lemma 4.9 to follow that the proof ultimately reduces to the case where V j v is distinguished. That is,

V j v = Y, V(2ki)

ki odd

and thus

E j u = Y, Jki,

ki odd

where for each i the Jordan block Jki can be taken such that Jki + Jkiv = V(2ki). This follows from the proof of Lemma 4.1 and the classification of outer unipotent elements.

In this situation, we choose a Levi subgroup L with respect to the given root system such that L' = n Aki-1. Let w0 be the long word in the Weyl group of G, and for each i let wi,0 be the long word in the Weyl group of Aki -1. Define

Y = wi,0. (6)

Then y induces a standard graph automorphism on each Aki-i-

For ki > 1, let Aki-1 have fundamental system ai,1,...,ai,ki-1, and let Xi = ui,1 • • • ui,{ki-1)/2, where ui,j is the root element Xi,,j(1). If v' = yx1x2 • • • , then (v')2 = u1u2 • • • , where each ui is a regular unipotent element in Aki-1. It follows that v' is conjugate to v and

so we can assume v = v' and u = u1u2 • • •in the following. Note that CAk._1 (v) ^ CG(v). Further, for each pair ki, kj, if we view Ak.-iAkj-i < Ak.+k,-i, then CAk.+kj_1 (v)° < CG(v)°.

Lemma 4.7. Assume V j v = £k. odd V(2ki) and let notation be as above. If ki > 1, then CaIi._1 (v) = U(ki-i)/2.2, with the component group generated by ui.

Proof. Simplify notation by considering Ar for r even and v = tu1 • • • u 2 squaring to a regular element u e Ar. Then setting u = v2, we see that E j u = J is a Jordan block of size r + 1 and the dimension formula (3) gives Cat (v)° = Ur. (This can be proved directly. Namely, CAr (u) is equal to Ur, an abelian group. The map $ : Ur ^ Ur sending d ^ dv = dT has its image in its kernel which is CUr (v). This implies that dim CAr (v) ^ |. The rest of the argument below will give the equality.)

Proceed by induction. The base case r = 2 is an easy calculation, so assume r > 2. Let Xp be the root subgroup corresponding to the highest root ¡3 = 11 • • • 11. Then Xp ^ CG(v) and v acts on P = NG(Xp) = QL, where L' = Ar-2, Q' = Xp, and Cq/q,(v) = (xii...iio(c)xoii...ii(c)Q' : c e K). However, (xii...iio(c)xoii...ii(c))v = xii...iio(c)xoii...ii(c)xiii...11(c), so that Cq(V) = Q'.

Now pass to P/Q where uQ = (vQ)2 is a regular unipotent element in LQ/Q = Ar-2. Inductively, Clq/q(vQ) = U(r-2)/2.2, with uQ generating the component group. Then CG(v)Q ^ Cp/q(vQ), so the assertion follows by induction and the previous paragraph. □

Lemma 4.8. Assume V j v = £k. odd V(2ki) is distinguished. Then CG(v) = CG(v)°(ui : ki > 1) and CG(v)/CG(v)° is abelian of exponent at most 2.

Proof. The assertion follows from Lemma 4.7 if V = V(2ki) with ki > 1. And if V = V(2), then G = 1 and again the assertion holds. So, now assume there is more than one summand and set W = Jkl, where Jkl + Jkl v = V(2k1). That is, Jkl is non-degenerate under ( , )v. Let D = Cgl(_e) (u) and M = WD. Then M = S U N, where N consists of the non-degenerate Jordan blocks of u on E and S consists of the blocks with non-zero radical. So, N is open and dense in M. As D is connected, M is irreducible.

We claim that C = CG(v) is transitive on M. Suppose J, J' e M and write E = J ± X = J' ± X'. Then R = J + Jv and R' = J' + J'v are non-degenerate in V of type V(2k1).

We next consider the action of v on R. To simplify notation for just this paragraph we will regard V = R = V(2k1). In the proof of Lemma 4.1, we gave a basis B = {v2kl,..., vj} with corresponding inner products, and the precise action of v and corresponding action of u on a pair of maximal singular spaces interchanged by v. One singular space is the Jordan block of u generated by v2kl; the other is its image under v. By [11, Theorem 4.2] the reductive part of Z = Csp(_R)(u)° is a 1-dimensional torus. The 1-dimensional tori of Z are all conjugate and each determines a unique pair of totally singular spaces. One such torus determines the pair J, Jv. Therefore, conjugating by an element g e Z, we can carry the above basis to the action of vg on R with J, Jv being the singular spaces interchanged by vg. Then there exists k e SL(J) such that v = vgk. Therefore, there is a basis Bgk of R such that the action of u on J, the action of v on R and the form on R are precisely as in the proof of Lemma 4.1.

Returning to the general case, a corresponding basis exists for R' = J' + J'v. Write V = (J + Jv) ± (X + Xv) and similarly V = (J' + J'v) ± (X' + X'v). Decompose X + Xv and X' + X'v and repeat the above argument. An application of Witt's theorem then yields an element y e CSp(y)(v) sending one basis to the next. In particular, y sends J to J' and also J + X to J' + X', and hence y e GZ, where Z is a torus inverted by v. As y centralizes v, we have y e C and the claim follows.

Now taking W G M, the orbit WC° is open in its closure in M. The closure must be M, as otherwise M would be a union of finitely many such orbit closures. Hence, WC is open dense in M. There are only finitely many such orbits (since C is transitive on M and C/C° is finite), so each must be closed as well. But this also contradicts irreducibility, unless C° is transitive on M. Hence, C° is transitive on M = WC and we can write C = C° StabC (W). Writing E = W .Lv Y, we have StabC (W) = CW x CY, where CW is the centralizer of v in SL(W) viewed as a subgroup of Sp(W + Wv) = Sp(2k1), and CY is the centralizer of v in SL(Y) < Sp(Y + Yv). Lemma 4.7 and induction now yield the result. □

Lemma 4.9. It suffices to establish Theorem 1.2 when v is distinguished in Sp(V).

Proof. By Lemmas 4.4-4.6

C/Ru(C) > n Sp2ai X n I2bi,

mi odd ni even

where ^ = O2bi or Sp2b..

Write V = Vc + Vd, where Vc j v = £ W(2mi)ai + £ W(2ni)bi and Vd = £ V(2ki). Let T0 be a maximal torus of C = CG(v), so that T0 projects to a maximal torus of C°/Ru(C) and a Frattini argument gives C = C°NC(T0). By Lemma 4.4 we can choose T0 to act trivially on Vd. Then Vd = [V, T0] so that NC (T0) = Nc X Nd, the induced actions on Vc and Vd, respectively.

Moreover, NC(T0) permutes the homogeneous components of the action of T0 on V. Each such component is invariant under the action of v, so NC (T0) permutes components on which v has Jordan blocks of a given size. It follows that NC(T0) preserves the decomposition Vc = ^ Vi, where each summand is one of the W(2mi)ai or W(2ni)bi summands and T0 is a corresponding product of subtori, each acting on a single summand (of dimension 4miai or 4ni6i).

For each i, the group NC(T0) acts on the maximal singular spaces Ei = Vi n E and Eiv, and we let Ni denote the restriction of Nd to the summand Vi. Then NC (T0) = Nc X Nd and Nc = EINi, where Ni = NCsl(e)(v)(T0). Also, Nd centralizes T0, since T0 acts trivially on Vd.

It follows from Lemma 4.4 that for each i we have Ni/Ru(Ni) = TiZ2ai Symai or TiZ2bi Symbi as appropriate, and that Ni/(Ni n CSL(Ei)(v))° has order at most 2. (It may happen that the order is 2 but Ni < C° due to Lemma 4.5.) Also Lemma 4.8 shows that Nd/Nd° is elementary abelian. Therefore, C/C° is elementary abelian. Since C°/Ru(C) is a product of symplectic and special orthogonal groups, each normalized by NC(T0), we have C/Ru(C) = rim- odd Sp2ai X n, even I2bi X Z, where Z is elementary abelian. A Frattini argument shows that Z is covered by the component group of CC (T0). Now, CC (T0) acts on Vc and Vd so that CC(T0) = Cc x Cd = Cc x Nd. Also CC(T0) n C° = T0 x CRu(C)(T0), which is connected. Moreover Cc is connected as it is contained in Ru(Nd)T0. It follows that Z is isomorphic to the component group of Nd which thus lifts faithfully to the component group of C. The result follows. □

In view of Lemma 4.9, we now assume V j v to be distinguished. Therefore, V j v = T,ki odd V(2ki) and E j u = £ki odd Jk-, where Jk- + Jk-v = V(2k) for each i. Then CG(v) < CSp(V)(v) is a unipotent group.

We first settle a few small cases which will be needed in inductive arguments to follow.

Lemma 4.10. Assume (G, u) = (SL4,J3 + J1), (SL6,J5 + J1) or (SL6,J3 + J3). Then CG(v)/CG(v)° = 1, Z2 or Z2, respectively. In the latter two cases the component group is generated by the image of u1, and in the last case u G CG(v)°.

Proof. In the first case we can consider SL4 (t) as O6. Then v is an outer unipotent element of order 8. Using [11, (4.4) and Theorem 4.2], we see that the component group of Co6 (v) is Z2 (generated by v), so CG(v) is connected. For the other two cases G = SL6 and we will determine the centralizers explicitly.

In the notation given in the paragraphs preceding Lemma 3.4, the labelled Dynkin diagram A corresponding to the Jordan decomposition J5 + J1 is 22022; let P be the parabolic subgroup determined by A. Set v = Tu3u2u1, so that u = v2 = u4u34u5u2u1. A direct check shows that u does have Jordan decomposition J5 + J1, so this forces V j v to be distinguished of type V (10) + V (2).

By Lemma 3.4, CG(v) ^ CG(u) ^ P. From the form of v it is immediate that Cp/q(v) < TGXa3Q and as CG(v) is unipotent, this forces CG(v) < U. Using the computer program described in the appendix we find that Cu(v) consists of all elements

xi (Z)x5 (Z)x2 (Z)x4 (Z)xi2 (t 1 )x45 (ti)x34 (Z)xm(t2)x345 (t 1 + ¿2 )

X x234(ti + ¿2)xi234(Zti + ti2 + ti + ¿2)x2345(t 12)x 12345(¿3),

where Z is in the prime field and t1, t2,t3 range over K. Working modulo Q, we have Cp/q(v) ^ TGU3, where U3 is a connected unipotent group of dimension 3. From this, it is easy to see that CG(v) ^ Q. From the expression above, we see that CG(v)° = U3 and this is the group obtained from taking Z = 0. Therefore, CG(v)/CG(v)° = Z2 as asserted.

For the last case the labelled Dynkin diagram A corresponding to the Jordan decomposition J3 + J3 is 02020; let P = QL be the parabolic subgroup determined by A. Let v = Tuooioouiiooouoiioo so that u = v2 = u3u45u34^ui2u23 = u45u345u34ui2u23 lies in Q. Note that x = uu5 = (u34u12)(u45u23) which is clearly of type J3 + J3. However, this does not by itself determine the class of v, since both types W(6) and V(6)2 have square in the class of u. This will be settled shortly. Here, we find that Cu (v) consists of all elements

xi(ti )x5 (ti )x2 (Z )x4 (Z )x3(ti)xi2(t2 )x45 (t2)x23(Zti + Z + t2)x34(Zti + t2) X xi23(t22 + t2)x345(t22)x234(t3)xi234(t4)x2345(Zt2 + t2 + t4)xi2345(t5),

where Z is in the prime field and t1,...,t5 range over K. Therefore, Cq(v)° = U4. If v had type W(6), then in view of the dimension information and Lemma 4.4 it would follow that CG(v)° = U4Sp2. Now CG(v) < CG(u) < P. Further, Cg(u)/Cq(u) = SL2. With x as above, x is centralized by the untwisted diagonal A1 = (x1(t)x3(t)x5(t),x-1(t)x-3(t)x-5(t) : t e K) in the Levi factor, so that u is centralized by this group conjugated by u5. However, v does not centralize this group modulo Q. Therefore, V j v is indeed distinguished of type V(6)2 and the dimension formula implies that CG(v)° = U5.

Now CG(v) = CP(v) normalizes CG(v)°Q/Q = (x1(c)x3(c)x5(c) : c e K)Q/Q. As CG(v) is unipotent and normalizes CG(v)°Q/Q it follows that CG(v) = Cu(v). Therefore, CG(v) has component group Z2, generated by u1 = u23u45. Also u e CG (v)°. □

The next two results will be used in certain inductive arguments.

Lemma 4.11. Assume that S < CE(u) is singular under ( , )v. Set E = S±v/S and V = (S + Sv)±/(S + Sv). Then S±v = E n (Sv)±, and the following hold.

(i) There is a natural embedding E <V as a maximal totally singular subspace (under the induced symplectic form) such that V = E + Ev.

(ii) The group PS = stabG(S) n stabG(S^) is a v-invariant parabolic subgroup of G which acts on each of V, E and Ev.

(iii) There is a v-invariant factor of PS/Ru(PS) which induces SL(E) on each of E and Ev.

Proof. For s G S and e G E,

(s,e)v = (s,ev ) = (su,ev) = (sv ,e) = (e,sv),

so that S^ = E n (SvTherefore, S±v = E n (S + Sv)± and this yields an embedding E < V. Taking images under v, we obtain (i).

From the previous paragraph, stabG(S±v) = stabG(E n (Sv= stabG(Sv). Therefore, stabG(S) n stabG(S±v) = stabG(S) n stabG(Sv) = PS is a v-invariant parabolic subgroup of G. Also PS acts on V and stabilizes E and Ev. Moreover, PS' has a v-invariant Levi factor inducing SL(E) on E. Parts (ii) and (iii) now follow. □

Lemma 4.12. Assume V j v = k- odd V(2ki) is distinguished and E j u = ^k- odd Jki, as above. Let Y ^ E be a sum of some of the blocks Jki with ki > 1 and let S = CY (u). Then S is singular in E with respect to ( , )v, and with notation as in Lemma 4.11 the following hold.

(i) The restriction E j u = ^- odd Jii (an orthogonal sum under ( , )v), where li = ki - 2 or ki, according to whether or not Jki ^ Y.

(ii) The restriction V j v = ^ V(2li) (an orthogonal sum, possibly not distinguished).

Proof. The summands in Y have the form Jki for ki > 1. It follows that for each i the fixed space of u on Jki + Jkiv = V(2ki) is singular, and as Y + Yv is an orthogonal sum of such spaces, S + Sv = CY+Y«(u) is singular in V, which means that S is singular in E with respect to ( , )v. The result now follows from Lemma 4.11. Note that V j v is distinguished if and only

if there do not exist distinct i, j, k with li = lj = lk, which by (i) may or may not be the case.

Lemma 4.13. Suppose k is odd, E [ u = Jk + Jk and V [ v = V(2k) + V(2k).

(i) If k = 1, then u e CG(v) = CG(v)°, a 1-dimensional unipotent group.

(ii) If k > 3, then u e CG(v)°. Also, CG(v) = U2k-1.2.

Proof. (i) Here G = SL2, so that v induces an inner automorphism of order 2 (as v is distinguished) and CG (v) is a 1-dimensional unipotent group. For (ii) consider Ak-1Ak-1 < A2k-1. With notation as in (6), we may write v = 7x1x2, where xi = ui1 • • • ui (k-1)/2 for i = 1, 2. Then v2 = u1 u2 where u1,u2 are regular unipotent elements in the corresponding Ak-1 factors.

There is an element s G CG(v) such that uj = u2j for each j. Then u1s = u2 so that [u1, s] = u1-1u2 = u mod CG(v)° by Lemma 4.7. Now Lemma 4.8 implies that u G CG(v)°.

The dimension of CG(v) is 2k - 1, by the formulae (3), (4) in Subsection 1.3. Hence, to complete the proof it suffices to show that u1 G CG(v)°. We proceed by induction. Lemma 4.10 gives the assertion for k = 3. Assume k > 3 and consider the parabolic subgroup P = stabG(CE(u)) n stabG(CE(u)±v). By Lemmas 4.11 and 4.12, P = QL is v-invariant and V j v = V(2k - 4) + V(2k - 4). Also, L' = A^k^! and uQ G A2k-5 has type J-2 + Jk-2 on the natural module E for A2k-5. Inductively, u1Q G CA2k-5Q/Q(v)°. On the other hand, Cg(v)°Q/Q < Clq/q(v)°, so this gives the assertion. □

Lemma 4.14. Suppose E [ u = Jk+2 + Jk with k odd. Then

(i) u e Cg(v)°;

(ii) if k > 1, then x1 e CG(v)°.

Proof. Proceed by induction. The base case E [ u = J3 + Ji is settled in Lemma 4.10, so assume k ^ 3. Here, CG(u) ^ P where P is determined by the labelled Dynkin diagram 2020 • • • 202 (see the discussion prior to Lemma 3.4). Then P is r-invariant, and Lemma 3.5 shows that this parabolic subgroup is also invariant under the action of v. Letting Q = Ru(P), we see that Z(Q) = Ua, a root subgroup. Then Co(Ua) = P = QL, a v-invariant parabolic subgroup containing P, with L' = A2k-i. So uQ = uiu2Q has type Jk2 on the natural module for A2k-i. Then dim CL' (vQ) = 2k — 1, whereas dimCG(v) = 2k.

Now Q/Q' has the structure of the sum of a natural module for L' and its dual, with the terms interchanged by v. Therefore, the fixed space of u has dimension 4 and the fixed space for v has dimension 2. Denote the latter space by F/Q'.

We claim that [v, F] = Q'. To see this consider just the factor Ak+1.2 and the element ru1. Using a similar argument to that in the first paragraph, we obtain a parabolic subgroup P = QL which is the normalizer of a root subgroup. Then Cq/q (v) consists of all elements of form X(c) = xii...ii0(c)x0ii...ii(c) for c G K (root elements relative to the Ak+1 system). As in the proof of Lemma 4.7, X(c)v = X(c)xii...ii(c). Since Q < Q, the claim follows.

By the claim, the map f ^ [v,f ] from F to Q' is surjective, so CF(v) has dimension 2. Moreover, F/Q' is the direct product of the groups (X(c) : c G K)Q'/Q' and CF(v)Q'/Q', so it follows that CF(v) is connected, whence Cq(v) = U2, a connected unipotent group of dimension 2.

Let X/Q = CG(v)°Q/Q and Y/Q = Cl>q/q(vQ). It follows that X/Q has codimension 1 in Y/Q and hence is normal. The proof of Lemma 4.13 shows that there is an element sQ G Cl'q/q(vQ) such that [sQ, uiQ]= uQ G Y/Q. As the component group of Clq/q(vQ) is generated by xiQ, we may take sQ G Y/Q. But Y/X is a 1-dimensional unipotent group and hence is centralized by xiQ. It follows that uQ G X/Q so that u G CG(v)°, proving (i). Part (ii) now follows using induction and Lemma 4.13. □

Lemma 4.15. Suppose E [ u = Jkl + Jk2 with ki > k2 +2. Then

Z2 x Z2 if k2 > 1,

CG(v)/CG(v)° =

Z2 if k2 = 1.

Proof. Proceed by induction. If k2 > 1, let

P = stabG(CB(u)) П stabG(C_E(u)±u),

which by Lemma 4.11 is a v-invariant parabolic subgroup. Then P = QL, where L' = AiAkl +k2_5Ai. Lemma 4.12 implies that uQ has type Jkl-2 + Jk2-2 on the natural module E for Akl+k2-5, and V I v = V(2ki — 4) + V(2k2 — 4). If k2 > 3, then inductively the component group is Z2 x Z2, so Lemma 4.8 implies that the same holds for C.

Suppose that k2 = 3. Then induction shows that the component group modulo Q is Z2, generated by uiQ, whereas u2Q is trivial. In particular, ui,u / CG(v)°. Let Pi = stabG(S) П stabG(S^"), where S is the fixed space of u on Jkl. Now consider the quotient S/S and repeat the process until one arrives at a Levi factor of type A5 where the image of u has type J3 + J3. It follows from Lemma 4.13 that the image of u2 is not contained in the connected centralizer of the image of v. Hence, u2 G CG(v)°, which together with the above yields the assertion.

So, now assume k2 = 1. Here, we must show the component group is Z2. Let Pi be as in the second paragraph. We again obtain the result inductively, provided ki > 5; otherwise, the hypothesis does not hold in the quotient space. So, we are reduced to the case E [ u = J5 + Ji, where Lemma 4.10 gives the assertion. □

We can now complete the analysis of the component group in the distinguished case, which, in view of Lemma 4.9, is all that is required.

Lemma 4.16. Theorem 1.2 holds if V I v = £ ki odd v№)•

Proof. Here, E j u = ^¿=1 Jki, where the ki are odd and in non-increasing order. There is an equivalence relation generated by the condition that ki and ki+1 are related (linked) if either ki = ki+1 or ki - ki+1 = 2.

Let C = CG(v). Then it follows from Lemmas 4.8 and 4.7 that C/C° is generated by commuting elements si = uiC° where ki > 1. Lemmas 4.13 and 4.14 show that sisi+1 = 1 if ki and ki+1 are linked. Also, if kr = 1, then any ki linked to 1 satisfies ui G C° by Lemmas 4.14 and 4.10.

If Ci is one of the equivalence classes (linkage classes), then ujum G C° if kj,km G Ci. So, we must show that a product of terms uj is in C° if and only if it has the form c1c2 ••• , where each ci is a product of an even number of terms uj for kj G Ci, or any product of terms uj if kj is linked to kr = 1.

We show by induction on dim V that there are no other relations. This will establish Theorem 1.2 for the case where V j v is distinguished. Suppose w = J} uij G C° with i1 > i2 > ••• where w is an element of minimal length not of the above form. In particular, none of the uij has the corresponding kij either equal to or linked to 1.

First assume kr > 1. Let R = CE (u) and P = stabG(R) n stabG(R±v). Then by Lemmas 4.11 and 4.12, P = QL is a v-invariant parabolic subgroup such that L' = ArAsAr, where r = dimR, s = dim(R±v/R) and uQ G AsQ has type ^r=1 Jki-2.

By induction we obtain a contradiction if there are at least two terms in the product and ki2 ^ 5. For the exceptional cases first assume there are just two terms and ki2 = 3. Then wQ = ui1 Q and by minimality ki1 - 2 is linked to 1, so that ki1 is linked to 3. That is ki1 and ki2 are linked, contrary to our hypothesis.

Now suppose w = uki. Here induction gives a contradiction unless ki - 2 is linked to 1, that is, ki is linked to 3. Consequently, we may assume ki = kr = 3. Let S = [E, u](5) n CE(u) and let P = stabG(S) n stabG(S v). This time uQ has the form y^k >7 Jki-2 where d is the multiplicity of J3 in E j u and c is the sum of the multiplicities of J7 and J5.

If c ^ 2, then v is distinguished on V and induction gives a contradiction. If c > 2, then

V I v V(2ki - 4) + V(10)c + V(6)d

= W(10) +Y V(2ki - 4) + V(10)c-2 + V(6)d.

The image of w comes from the V(6)d summand, within the distinguished part of the sum. The argument at the end of the proof of Lemma 4.9 shows that the component group of the distinguished part lifts faithfully to the full component group. Inductively, we again conclude that ukiQ is not in the connected centralizer of v in P/Q.

Finally, we return to the previously excluded case kr = 1. The above shows that w only involves terms uij for kij ^ 5. Let S = [E, u] n CE(u) and let P = stabG(S) n stabG(S±v), as in Lemma 4.11. Then V = ^2k->5 V(2ki - 4) + V(2)c, where c is the sum of the multiplicities of V(6) and V(2) in the expression for V j v. If c ^ 2, then inductively wQ is not in the connected centralizer of v in P/Q, so w / C°. And if c > 2, then F = W(2) + J2k->5 V(2ki -4) + V(2)c-2. But as w only involves terms ui. for ki. ^ 5, it follows that wQ only acts on J2k.>5 V(2ki - 4). We obtain a contradiction as in the last paragraph. □

4.5. Proof of Theorem 1.3

Recall that G = SLn(K). As in Subsection 1.1, a is a q-field morphism commuting with t, and Y is either a or aT, with GY = SLn(q) or SUn(q), respectively. Again, we regard G(t) as a subgroup of Sp(V), where V = V2n (K) = E © ET and G acts on each of the maximal totally singular summands E and ET. Given v e Gt, Equation (1) in Subsection 1.3 states that

V j v = Ys W(2mi)a* + Y W(2m)bi + Y V(2ki).

m* odd n* even k* odd

We first claim that a representative for the G-class of v can be written over the prime field. To see this it will suffice to show that this holds for the individual summands W(2m) (m even or odd) and V(2k) (k odd).

The proof of Lemma 4.1 shows that Sp2k(2) contains an element v acting on the symplectic module as V(2k), and for k odd this element interchanges two singular subspaces. This settles the V(2k) case. Now consider the W(2m) case. For m odd consider Sp2m(2) x Sp2(2) acting on V2m ® V2 and take v to be a regular element in the first factor. Then v2 acts as Jm + Jm' on the V2m space, with v interchanging the blocks. Hence, v interchanges the singular spaces Jm ® V2 and Jm' ® V2, giving the assertion. Finally, for m even, set v = Tu1u2 • • • um-1. Then v2 has Jordan form Jm2 on E. As an element of Sp(4m), we could have V j v = W(2m) or V(2m)2. But since v e SL2mt the latter is impossible, as is shown by the decomposition (1) repeated above. This establishes the claim, from which it follows that Y stabilizes each orbit in the action of G on outer unipotent elements in Gt .

Let v e SL2m(2)T be as above and set C = CG(v). By the above we can choose a and hence y to normalize each of the factors Gi = SL2ami, SL2b*n* and SLki corresponding to the decomposition (1). Now consider the action of y on CG(v). Using Lemmas 4.4 and 4.7, we see that y acts on the appropriate classical group CG*(v)/Ru(CG*(v)), centralizing the component group. This implies that y leaves invariant each of the factors Sp2a* and I2b* (even if a group O2b* pumps up to Sp2b.) of C/Ru(C) and acts trivially on C/C°.

At this point, we apply the usual Lang-Steinberg theory; we refer the reader to [10] for details. We find that vG n GYt splits into 2s+t+5 classes and these correspond to representatives cC° of C/C°. For such a representative consider the fixed points of yc (a G-conjugate of y) on C. Setting D = Ru(C) = Ud, we see that \DYC\ = qd and Cyc

covers (C/D)yc. Moreover,

Yc acts on each of the factors Sp2a. or I2b* as a field or graph field morphism, with all 2s possibilities for the fixed points occurring equally often. Theorem 1.3 follows.

5. Proof of Theorems 1.4 and 1.5

In this section, we assume (G,p) = (E6, 2) or (D4, 3) (with G simply connected or adjoint in the latter case); we shall prove Theorems 1.4 and 1.5.

5.1. Possibilities for the pth power u

We first seek to determine the possible unipotent elements u that can arise as the pth power of an element v e Gt. At this stage, we shall obtain a list of candidate elements; later we shall see which of these possibilities actually occur. For convenience, in this subsection we shall take G to be simply connected.

As mentioned in Subsection 1.1, we let H be a simply connected group of type E7 or F4 according as (G,p) = (E6, 2) or (D4, 3); the assumption on the isogeny type of G means that we may regard G(t) as a subgroup of H. Indeed, in the former case, H has a Levi subgroup E6T1 with normalizer (E6T1).2, in which an outer involution induces a graph automorphism

of Ee and inverts T1; in the latter case, the subgroup of H generated by all root subgroups corresponding to long roots is D4, with normalizer D4.S3.

Recall that for X = G or H, we denote by VX (Ai) the restricted irreducible X-module with high weight equal to the ith fundamental weight. In particular, VE?(A7) is the restricted 56-dimensional module for E7. For p = 3, we will denote by WF4 (A4) the 26-dimensional Weyl module for F4 with high weight A4, which has the 25-dimensional irreducible module Vp4(A4) as a quotient. As before, we write Ji for a Jordan block of size i; we consider the action of v G Gt and u = vp G G on certain modules for H or G.

Lemma 5.1. With notation as above, assume vp = u.

(i) If (G,p) = (Ee, 2) and VEfi(A1) j u = J„i + ••• + Jat, then VEr(A7) j v = .hai + ••• + J2at + J2.

(ii) If (G,p) = (D4, 3) and VDAA1) j u = Jai + ••• + Jat, then WFi(A4) j v = J3ai + ••• + J3at + J2 or J3ai +-----+ J3at + J12.

Proof, (i) We have VEr (A7) j Ee = VEe (A1) © VEe (Ae) © V2, where V2 is a 2-dimensional space on which Ee acts trivially. Under the action of the 1-dimensional torus T1 mentioned above, the space V2 decomposes as a sum of two weight spaces for distinct weights. Therefore, t interchanges the modules VE6 (A1) and VE6 (Ae) and also the weight spaces of T1 on V2.

If J is a Jordan block of u on VE6 (A1), then Jv is a Jordan block of u on VE6 (Ae) and J + Jv is invariant under v. Further, the fixed space of v on J + Jv is 1-dimensional, from which it follows that J + Jv is a single Jordan block of v. Also, v acts on V2 as a single Jordan block. The assertion follows.

(ii) Here, Wf4(A4) j D4 = Vd4(A1) © Vd4(A3) © Vd4(A4) © V2, where V2 is a 2-dimensional space on which D4 acts trivially. The above argument gives the assertion, noting the ambiguity for the action of v on V2. □

In [9], the first author gives the Jordan structure of unipotent elements of Ee on VE6 (A1), of E7 on VE7 (A7) and of F4 on WF4 (A4). Using this together with the known Jordan structure of unipotent elements of D4 on VD4 (A1) described in the proof of Lemma 3.2(iii), we may employ Lemma 5.1 to obtain the list of possibilities for the H-class containing v and the G-class containing u. The notation is as in [11].

Lemma 5.2. Tables 1 and 2 list the possibilities for the H-class of v and the G-class of u = vp which are consistent with the above information on Jordan block sizes.

In the first column of Tables 1 and 2, for each possible G-class we give the corresponding labelled Dynkin diagram A; recall from Section 3 that A determines the parabolic subgroup P = QL of G, where the simple roots of L are those with label 0 in A. Note that each such P is T-stable; by Lemma 3.5 it follows that each possible v stabilizes the corresponding P.

We conclude this subsection by providing, for each possible G-class uG, a precise expression for a representative u in the form upi.

Lemma 5.3. For each of the G-classes uG in the second columns of Tables 1 and 2, an explicit representative u is given in Tables 3 and 4.

Proof. In most cases, it is clear that the product of unipotent elements given is in the correct class. For example, consider the expression for the elements of type A2A12 or A4 in Table 3. Each of these has the form u = uau@uYug. The roots a, ¡3, y, 5 form a simple system for a root system of type A2 A12 or A4, as appropriate. Then u projects to a regular element in

TABLE 1. Possible classes uG and vH for G = E6.

A uG vH

00000 0 1 A14, (A13)"

00000 1 Ai A2A13

10001 0 A12 A3A12, (A3A1)"

00000 2 A2 D4A1

00100 0 A13 D4(0,1)A1, A3A2A1, (A3A2)2

20002 0 A22 AB",D6(o2),ABA1

20002 2 A4 D6

01010 0 A2A12 D5(o1)A1

00200 0 D4(«1) D5A1

10101 0 A22A1 Et(05)

20202 0 E6(as) E7(o2)

22022 2 Ea(«1) e7

TABLE 2. Possible classes uG and vH for G = D4.

A uG vH

000 10 1 20 2 1 A13 D4(o1) A2,A2A1 C3,F4(02) F4

the corresponding subsystem subgroup. A similar analysis covers all cases other than Eg(ai), E6(a3) and D4(ai). For each of these cases let u be the element given.

Consider E6(a1). The labelled Dynkin diagram is 2 , so that u G Q = Q^2. To see that u is in the correct class it suffices by parts (ii) and (iii) of Lemma 3.4 to show that uQ>2 is in the dense orbit of L on Q/Q>2. Now P is a distinguished parabolic subgroup, so dimL = dim(Q/Q>2), and it will suffice to show that the stabilizer in L of uQ>2 is finite. The results of [2] imply that L' acts on Q/Q>2 as on the sum of two trivial modules and three natural modules. Consider the projections of uQ>2 to the modules (U21222 U21122 )Q>2 and

(U22122 U22112 )Q>2. The projections are, respectively, minimal and maximal vectors of these

natural modules. As CL(uQ>2) must stabilize each of these projections, we conclude that CL(uQ>2) ^ Tg, a maximal torus of L. But as the roots appearing in u and their negatives generate the full root system of G, we conclude that CL(uQ>2) = 1.

Similar but easier considerations apply to D4(a1) in Table 3. Start with the subsystem subgroup of type D4 with simple system 11111, 22222, 22122, 21212. Then u is contained in the unipotent radical of the parabolic subgroup determined by the labelled Dynkin diagram 22 22 , and as above we see that u is distinguished in this D4. Now D4 has two conjugacy classes of distinguished unipotent elements, namely the regular elements and those acting on the usual orthogonal module as the sum of two orthogonal Jordan blocks of size 4. Clearly, u is not a regular element since it lies in all Borel subgroups of the parabolic subgroup indicated. So, u has type D4(a1). The case of the class D4(a1) in Table 4 is similar.

ROSS LAWTHER, MARTIN W. LIEBECK AND GARY M. SEITZ Table 3. Class representatives u in G = E6.

A2 Ai2

D4(ai) A22Ai Еб(аз) Ee(ai)

u12321 2

u12211u11221 ii

u01210 u11111 11

u11211 u12210u01221 111

u00111u01111u11100 u11110 1010

u00011 u00110u11100u01100 0101

u01110u11111 u11210u01211 0111

u00100u11111 u00100u01110 1100

u00111u01111u11100 u11110 u01210 10101

u11000u00011 u01110u10000u00001 u00100 000001

u00001 u00100u00110u10000u00000u01000 010010

TABLE 4. Class representatives u in G = D4.

D4(a1)

u-,-1 1 u11 0 u01 1

110 111 011

^000 u00 0 u011 u10 0 u110

Now consider E6(a3). As noted in the proof of Lemma 3.2(ii), elements of this type are represented in F4 as unipotent elements of type F4(a2). Also uT = u,so u G F4. With u = П upi, set e = 5^ epi. Then from the F4(a2) nilpotent element of [11, Table 13.3] and the usual folding of the root system, we see that in the Lie algebra L(F4) the nilpotent element e is distinguished of type F4(a2). Now [11, Lemma 19.7] shows that u is distinguished of type F4(a2) in F4 and hence is a distinguished unipotent element of type E6(a3) in G. □

5.2. The elements Vi

In this subsection, we shall consider the elements vi listed in Tables 7 and 8, and begin the process of showing that the information on each element presented there is correct. We continue to assume G is simply connected, so that G(r) < H, where H is simply connected of type E7 or F4.

We will require the following standard notation. Recall that in the root system E we have the simple system П = {ai,. ..,ar}. For j = 1,...,r and c G K*, let hj(c) denote the usual element of (Xa., X-a.) П TG such that xa. (t)hj(c) = xa. (c2t) for all t G K. Explicit expressions for these elements are given in [3, Lemma 6.4.4], although adjustments must be made to account for the fact that here we are acting on the right rather than the left.

It will also be convenient to use a certain abbreviated notation. As examples set

хз(с) = x01000 (c), X245 (с) = X 00110 (c),

x3,5(c) = x01000 (c)x00010 (c), xi,4,6(c) = x 10000 (c)x00000 (c)x00001 (c),

etc. Similarly, set

Y2 = x (c),x-2 (c)), Y2,3,5 = (x2 (c)x3(c)x5(c),x-2(c)x-3(c)x-5(c)), Y13,56 = (x13,56(c),x_13i_56 (c)),

Finally, we give r explicitly as an element of H. For ¡3 a root of H we write s^ for the standard representative of the Weyl group reflection in ¡. According as H = E7 or F4, we take

r = S 122111 s 112211 s 012221 or r = S0001S0010-111

In the former case this suffices to determine the action of r on G. In the latter case, however, in order to distinguish r from its inverse we must specify the correspondence between roots of G and of H. We take 10 00 , 01 00 , 00 01 and 001 to be 0100, 1000, 0120 and 0122, respectively; thus, with r as above we have

-y T _ -y V T _ -y V T _ -y

A10° = A00 0 , A00 0 = A001 , A001 = A100 •

Our first result here is then the following.

Lemma 5.4. For each element vi listed in the second column of Table 7 or 8, its pth power vip is the element u listed in Table 3 or 4 for the corresponding G-class.

Proof. This is simply a direct check. □

We next determine the H-class of each element vi.

Lemma 5.5.

(i) For G = E6 and i = 1,---, 17, the E7-class of the element vi is as indicated in the third column of Table 7.

(ii) For G = D4 and i = 1,---, 5, the F4-class of the element vi is as indicated in the third column of Table 8.

Proof. (i) A computer calculation determines the Jordan forms of the elements vi on both Ve7 (A7) and L(E7). At this point, the results in [9] suffice to identify the class of vi, with the exceptions of v10 and v11. The Jordan form information shows that these particular elements must have type A5A1 or D6(a2), but these classes are not distinguished by their Jordan form on either VE7 (A7) or L(E7). However, it follows that each of v10 and v11 is centralized by a 1-dimensional torus, say T1, of E7 and is a distinguished unipotent element in the semisimple part of Ce7 (T1).

Now v11 is centralized by the 1-dimensional torus S = {^4(c) : c G K*}. Then S lies in a fundamental A1 subgroup, and it follows that CE7 (S) = D6S. Therefore, S is conjugate to T1 and v11 is distinguished in D6, and hence has type D6(a2) by the above.

Next v10 is centralized by the 1-dimensional torus S = {h2(c)h3(c)h5(c) : c G K*}. One checks that S centralizes the subsystem subgroup of type A5A1, where the A5 has simple

Table 5. CQ(vt) and Cu(vt) for G = E6.

A Vi cQ (Vt) cu (Vt)

00000 0 vi = T 1 U24

V2 = TU 12321 2 1 U24

00000 1 V3 = tu 01210 u 11111 11 U14 U20

10001 0 V4 = TU 12211 1 U15 U21

V5 = TU01210 u 12211 11 U11 U20

V6 = tu 01100 u 00110 u 01210 u 11111 u 12211 11111 U11 U20

00000 2 V7 = TS1S6S4U11111 1 V8 = TU 00100 u 11111 U 12210 101 U10 U12

00100 0 U15 U16

20002 0 V9 = TU11100 u 11110 10 U8 U14

V10 = tu 11100 u 11110 u 01210 101 U8 U14

V11 = TU01110 u01100 u00110 u 11100 u 11110 01110 U8.2 U12.2

20002 2 V12 = TS4U 11100 u 01100 01 U7.2 U8.2

V13 = TS4U 11100 u 01100 u 01110 010 U7 U8

00200 0 V14 = TS1S6S2U 00100 U 01110 00 U9 U9

10101 0 V15 = tu 11100 u 11110 u 01110 u 00100 1010 U12.4 U12.4

20202 0 V16 = TS2S3S5U10000u00001u00100 001 U6 U6

22022 2 V17 = TS4U 10000 u 00000 u 01000 010 U4.2 U4.2

Table 6. Cq(vt) and Cu (vt) for G = D4.

A Vi CqV) cu (Vt)

00 0 V1 = T 1 U6

V2 = TU12 1 1 U6

10 1 V3 = TU. , 1 U3 11 0 U4

V4 = tu1 -,1 U01 0 U3 110 010 U4.3

20 2 V5 = tu100 u01 0 U2 10 0 01 0 U2

system 11i000, 010110, 008001, 001110, 110100 and the A1 has simple system 011100. It follows that v10 has type A5A1 in E7.

(ii) As above, the result follows from calculating the Jordan forms of the elements vi on both L(F4) and Wp4 (A4) and applying the results of [9].

5.3. The centralizers CG(vi)

By now the only entries in Tables 7 and 8 which we must establish are those in the final column, giving the centralizers CG(vi). Recall that by the final paragraph of Subsection 1.1 these are independent of the isogeny type of G; we shall in fact assume G is adjoint in this subsection.

In the lemmas to follow we shall make frequent use of two pieces of information for a given element vi with pth power u. Firstly, the structure of CP(u)/Cq(u) is given in [11, Table 22.1.3]

or [11, Table 8.5a] according as G = E6 or D4. Secondly, Tables 5 and 6 give for each vi the structure of CQ(vi) and Cu(vi) (recall that U = peS+ Xp, where £+ is the positive system determined by n). The information in these tables summarizes results obtained by performing computer calculations to identify the U-centralizers explicitly; these results are presented in more detailed form in the appendix. For convenience of reference, the first column of Tables 5 and 6 gives the labelled Dynkin diagram A, which determines the parabolic subgroup P, while the second column repeats the definition of the element vi.

In the lemmas which follow, we treat together elements vi having the same pth power. We begin with G = E6.

Lemma 5.6. If G = E6, then CG(vi) is as given in Table 7 for i = 1, 2.

Proof. This is well known: it is shown in [1, 19.9] that CG(v1) = F4 and CG(v2) is isomorphic to the centralizer in F4 of a long root element. □

Lemma 5.7. If G = E6, then Gg(ví) is as given in Table 7 for i = 3.

Proof. Since v3Q/Q = rQ/Q, and CP(v3) < CP(u) = QA5 by [11, Table 22.1.3], it follows that CP(v3)Q/Q ^ C3Q/Q where the group C3 consists of the fixed points of r on the A5 Levi subgroup. If we write 51,52, 53 for the simple roots of this C3, then for c G K we have x^ (c) = x1(c)x6(c), xg2 (c) = x3(c)x5(c) and xg3 (c) = x4(c). We now produce a subgroup G2 in this C3.

Start with a group B3 defined over K, with simple roots ¡1,32, ¡3 (numbered in the usual manner). By taking the fixed points of a triality automorphism of D4, we see that there is a group G2 < B3, with simple roots a (short) and b (long), generated by root elements xa(c) = (c)x@3 (c) and xb(c) = x@2 (c) for c G K along with corresponding elements for negative roots. Since p = 2, we have a surjection B3 ^ C3 with (c) ^ x^ (c), x@2 (c) ^ xg2 (c), x@3 (c) ^ xg3 (c2). Thus, we have G2 < C3 generated by elements x^ (c)xg3 (c2) = x1(c)x6(c)x4(c2) and xs2 (c) = x3(c)x5(c) for c G K together with negatives.

Now by inspection of the detailed information given in the appendix for Cu(v3), we see that CP(v3) covers the maximal unipotent subgroup of this G2. The Weyl group of the G2 is generated by the involutions s1s6s4 and s3s5 of A5. One checks that s3s5 and s1s6s4u01210 centralize v3, so it follows that CP(v3)Q/Q contains a subgroup isomorphic to

G2. Now G2 is a maximal subgroup of C3, so if the containment were proper, we would have CP(v3)Q/Q = C3, contrary to Table 5 which states that Cq(v3) = U14 while Cu(v3) = U20. Therefore, CP(v3)Q/Q = G2, and thus CP(v3) = U14G2 as required. □

Lemma 5.8. If G = E6, then Gg(v¿) is as given in Table 7 for i = 4, 5,6.

Proof. Here, Q^2 = Z(Q) affords a natural orthogonal module for L' = D4, with u G Q^2 a non-singular vector. Then CP(u) < QB3T\ by [11, Table 22.1.3], where B3 = CL/(t) and Tí = {^i(c2)^3(c)^5(c-1)he(c-2) : c G K*}. We note that acts trivially on the orthogonal module and is inverted by t .

First consider v4 = tu 12211. Now B3 contains a subgroup Ü6 = D3.2 = (Y4,Y2,Y345)(s3s5),

where the D3 centralizes the non-degenerate 2-space of Z(Q) spanned by the root elements

u 12211 and u 11221, while s3s5 interchanges the basis elements. Also, as s3s5u 12211 centralizes 1 1 1 v4 we have CP(v3)/Cq(v3) ^ D3.2. As is a maximal subgroup of B3, Table 5 implies that this containment must be an equality, which yields the result for v4.

Now consider v5 and ve. Modulo Q these elements have the form v4x and v4y, where x is a long root element of B3 for the highest root, and y is the product of x and a short root element for the highest short root. Both x and y are central in the standard maximal unipotent subgroup, and it follows (see for example [11, Lemma 2.4]) that CB3 (x) and CB3 (y) contain derived groups of parabolic subgroups of B3. Checking fundamental reflections, we see that CB3 (x) = U7A1A1 while CB3 (y) = U8A1, where in the second case the A1 corresponds to the fundamental short root of B3.

Now CP(vi)Q/Q is contained in CB3 (x)Q/Q or CB3 (y)Q/Q, respectively, and the information on v5 and ve in Table 5 shows that CP(vi)Q/Q contains UQ/Q, where U is the standard maximal unipotent subgroup of B3. We see that CP(v5) contains s2 and s3s5u 12211,

whereas CP(ve) contains s3s5. It follows that CP(v5)Q/Q covers CB3 (x) while CP(ve)Q/Q covers CB3 (y). These are the derived groups of the standard parabolic subgroups with Levi subgroups Y2 x Y3,5 and Y3,5, respectively. The result follows. □

Lemma 5.9. If G = Ee, then CG(vi) is as given in Table 7 for i = 7.

Proof. Here, Cp(u)Q/Q = A2A2.2 by [11, Table 22.1.3], where A2A2 = (Ys^e^Y^) with the factors interchanged by s1ses4. The factors are also interchanged by t, so that v7Q = Ts1ses4Q acts as a graph automorphism on each A2 factor of A2A2Q/Q. It follows that Cp(v7)Q/Q ^ A1A1.2. Moreover, Y345e x Y1345 = A1A1 centralizes v7, as does s1ses4u01210.

Therefore, CP(v7)Q/Q = A1A1.2 and the conclusion follows using Table 5. □

Lemma 5.10. If G = Ee, then CG(vi) is as given in Table 7 for i = 8.

Proof. Here, CP(u)Q = A2A1Q by [11, Table 22.1.3] where A1 = Y2 and A2 = (Y3,e,Y1j5). Now, v8Q = tQ induces a graph automorphism on the A2 factor, so that CP(v8)Q/Q ^ A1A1, where A1A1 = Y2 x Y13 5e. The detailed information in the appendix shows that Cp(v8)Q contains the elements x13(c)x5e(c)x2(c2)Q for c G K. Also, v8 is centralized by s1S3 seS5 s2u00100 u01121. Therefore, CP(v8)Q/Q contains a diagonal A1 in A1A1. By Table 5

we must then have CP(v8)Q/Q = A1, and the result follows. □

Lemma 5.11. If G = Ee, then CG(vi) is as given in Table 7 for i = 9,10,11.

Proof. Here, u = u 00111 u 01111 u 11100 u 11110 . We have CP(u)Q/Q = G2 by [11, Table

22.1.3], and indeed G2 = (Y4,Y2,3,5) centralizes u. As this group also centralizes v9, we obtain CP (v9) = U8G2 from Table 5.

Now consider v10 and v11. These elements have the form v9x and v9y, where x is a long root element of G2 for the highest root, and y is a short root element for the highest short root. It follows that CP(vi)Q/Q is contained in CG2 (vi)Q/Q = U5A1 or U3A1, respectively. Here, the A1 factor is just Y2 3 5 or Y4 according as i = 10 or 11. Using the information in the appendix for these elements we see that CP(vi)Q/Q = Ue or U4 respectively. One checks that CP(v10) contains s2s3s5 while CP(v11) contains s4. It follows that CP(vi)Q/Q = U5A1 or U3 A1 respectively.

Another appeal to Table 5 shows that Cq(v10) = U8, whereas Cq(v11) = U8.2. This completes the analysis of CG(v10) = CP(v10), but for v11 we must verify that CP(v11) = U11A1.2. That is, we must verify that the component group of the centralizer is non-trivial. From the description of Cu (v11), it is clear that this group is disconnected with component group of order

2. Also the element s4 centralizes this component group. It follows that CP(v11) = U11A1-2, completing the proof. □

Lemma 5.12. If G = E6 then CG(vi) is as given in Table 7 for i = 12,13.

Proof. Here, we have u = u 00011 u 00110 u 11100 u 01100 , V12 = rs^u 11100 u 01100 and V13 =

0101 01

v12u01110. Further CP(u)Q/Q = A1T1 by [11, Table 22.1.3], where A1 = Y345 and T1 =

{h^c"2)^^"1)^^3)h5(c)h6(c2) : c G K*}. Note that ts4 inverts T1.

Now viQ = ts4Q or rs4u 01110 Q, according as i = 12 or i = 13. As ts4 centralizes A1, we have

Cp(V12)Q/Q < A1Q/Q and Cp(v13)Q/Q < X01110 Q/Q. By inspection, we have A1 < Cp(V12)

and X01110 ^ CP(v13). Therefore, the containments are equalities.

The result now follows from Table 5, except that we must determine the component group of CP(v12). However, we have seen that this group is the semidirect product of Cq(v12) and A1, and this implies that the component group of CP (v12) is just that of Cu (v12), which has order 2. □

Lemma 5.13. If G = E6, then CG(vi) is as given in Table 7 for i = 14.

Proof. Here u = u 00100 u 11111 u 00100 u 01110 , vM = rs1S6S2u00100 u 01110 , and Cp(u)Q/Q = 1 1 0 0 0 0

T2-S3 by [11, Table 22.1.3]. In this instance, we have T2 = {h1(a)h3(b)h5(b-1)h6(a-1) : a,b G K*}; this is inverted by r and CT2 (rs1 s6s2) is the 1-dimensional torus T1 = {h1(c)h6(c-1) : c G K*}. Also, r centralizes u and [11, Table 22.1.4] implies that r centralizes the S3 quotient of CP(u)Q/Q. It follows that Cp/q(v14Q) = (s1s6s2)T1Q/Q. One checks that T1 centralizes v14. Also r centralizes v14, and therefore so does rv14 G s1s6s2Q. Thus, CP(v14Q)/Q = (s1s6s2)T1Q/Q, and the result follows from Table 5. □

Lemma 5.14. If G = E6, then CG(vi) is as given in Table 7 for i = 15.

Proof. Here u has type A22A1 and CP(u)/Cq(u) = A1 by [11, Table 22.1.3]. Consider the group A = Y2,3,5, which is of type A1. Write h(c) = h2(c)h3(c)h5(c) for c G K*, and set T1 = {h(c) : c G K*}; then T1 is a 1-dimensional torus of A. Take w G K* with w3 = 1 = w. One checks that h(w) and s = s2s3s5u00100 centralize v15. Modulo Q these elements generate

a group of type S3. Therefore, CP(v15)Q/Q contains S3.

We claim that CP(v15)Q/Q = S3. To see this view A as a short root A1 in the group G2 < D4, where the D4 has simple system 01000, 00000, 00000, °0°10 and G2 is the group of fixed points under the standard triality automorphism. Consider the standard parabolic subgroup

P = QL > P, where L' is this group D4. Then v15Q/Q = ru01110u00100 Q/Q and we view this

as contained in (r) x G2. Now u01110u°°°00Q/Q is a unipotent element of type A2 in D4,

hence of type G2(a1) in G2. It follows from [11, Table 22.1.5] that the reductive part of the centralizer for this unipotent element is S3. This establishes the claim.

The appendix contains a precise description of the elements in Cu (v15). We conclude from this information that the component group of Cq (v15) is isomorphic to F4 and that h(w) acts non-trivially on this component group. Moreover, we see that x2345(c)x4 (c) G Cq (v15) for all c G K, which implies that s2 = u01110u00100 G Cq(v15)°. It now follows that the component

group of CP(v15) is isomorphic to S4, and so from Table 5 we have CP(v15) = U12.S4 as required. □

Lemma 5.15. If G = E6 then Ca(vi) is as given in Table 7 for i = 16.

Proof. Here u is a distinguished unipotent element of G and CP(u)Q/Q = Z2 by [11, Table 22.1.3]. As t centralizes v06, so does rvi6 G s2s3s5Q. It follows that CP(v16)Q/Q = Z2, so Table 5 gives CP(vi6) = U6.2 as required. □

Lemma 5.16. If G = E6, then CG(vi) is as given in Table 7 for i = 17.

Proof. Here u is a distinguished unipotent element of G and CP(u)Q/Q = 1 by [11, Table 22.1.3]; thus Table 5 gives CP(vi7) = Cq(v17) = U4.2 as required. □

Finally, we turn to G = D4.

Lemma 5.17. If G = D4, then CG(vi) is as given in Table 8 for i = 1,2.

Proof. This is well known: [7, Proposition 4.9.2] shows that CG(vo) = G2 and CG(v2) is the centralizer of a long root element in G2. □

Lemma 5.18. If G = D4, then CG(vi) is as given in Table 8 for i = 3,4.

Proof. Here, u = u11i u11ouQ1i has type Ao3, and CP(u)/Cq(u) = A1 by [11, Table 8.5a],

where Ao = Y"2. Table 6 shows that Cu(v3) = U4 and Cu(v4) = U4.3. Since Y"2 centralizes v3, we have CP(v3) = U3Ao as required. Now consider v4. Since v4Q = tuq1i Q, we have Cp/q(v4Q) =

(h2(- 1)}X(11i1 Q. We then see from the precise information on centralizers in the appendix that

CP(v4) = U4.S3 as required. □

Lemma 5.19. If G = D4 then CG(vi) is as given in Table 8 for i = 5.

Proof. Here u is distinguished in G with CP(u) = CQ(u) by [11, Table 8.5a]; so Table 6 shows that CP(v5) = U2 as required. □

5.4. Completion of proofs

We can now complete the proof of Theorems 1.4 and 1.5. We have shown that the information in Tables 7 and 8 is correct. Since the entries in the fourth columns of these tables are all different, it is clear that the elements vi represent distinct conjugacy classes in G(t}. What remains is to show that the vi form a complete set of conjugacy class representatives for the outer unipotent classes in Gt, and to verify that the information provided in Tables 9 and 10 is correct. An elementary calculation gives an alternative proof of Theorem 1.1 for these cases. In this subsection, we shall continue to assume G is adjoint.

Recall that a is the g-field morphism of G satisfying xp(c)a = xp(cq) for all ¡3 G £ and c G K, and y is either a or aT. From the expressions for the vi in Tables 7 and 8 we see

TABLE 7. Unipotent classes in Gt = E6t.

(Vi2)G Vi ViE7 Ca(Vi)

1 V1 = T (A13)" F4

V2 = tu 12321 2 A14 U15C3

A1 V3 = TU01210 u 11111 11 A2A13 U14G2

2 V4 = TU12211 1 (A3A1)" U15D3.2

V5 = TU01210 u 12211 11 A3A12 U18A1A1

V6 = tu 01100 u 00110 u 01210 u 11111 u 12211 11111 A3A12 U19A1

A2 V7 = TS1S6 S4U 11111 1 D4A1 U10A1A1.2

A13 V8 = tu 00100 u 11111 u 12210 101 A3A2A1 U15A1

A22 V9 = TU11100 u 11110 10 (A5)" U8G2

V10 = TU11100 u 11110 u 01210 101 A5A1 U13A1

V11 = TU 01110 u 01100 u 00110 u 11100 u 11110 01110 D6(a2) U11 A1.2

A4 V12 = TS4U 11100 u 01100 01 D6 U7A1.2

V13 = TS4U11100u01100u 01110 010 D6 U8

D4(a1) V14 = TS1S6S2U 00100 u 01110 00 D5A1 U9T1.2

A22A1 V15 = TU11100 u 11110 u 01110 u 00100 1010 Er(aB) U12.S4

Ea(«3) V16 = TS2S3S5U10000 u 00001 u 00100 001 E7(a,2) U6.2

Ea(«1) V17 = TS4U 10000 u 00000 u 01000 n 1 n e7 U4.2

TABLE 8. Unipotent classes in Gt = D4t .

(vi2)G

ViF4 Ca(Vi)

a42 G2

A2A1 U5 A1

C3 U3 A1

F4(a2) U4.S3

A13 D4(ai)

vi = T V2 = T«12 1

V3 = tu, 1 11 0

V4 = tu,, i u0 , o 11o o10

V5 = tu10o u01 o 1o0 o10

that a stabilizes each vi and hence the corresponding orbit Oi = viG. As vi = gr for some g G G, we have Oi = OiVi = OiT, and so t also stabilizes Oi for each i. Therefore, 7 stabilizes each Oi.

We now apply the usual Lang-Steinberg method to fi = Oi U ••• U On, where we set n to be 17 or 5 according as G = E6 or D4. For each i ^ n, we find that 7 fixes an element ji in Oi, so that ji G Gyt; moreover, Oi n GYt is a union of GY-orbits, and the number of orbits and their sizes are determined by the action of 7 on the component group of CG(ji) = CG(vi). We see from Tables 7 and 8 that this component group is either 1, Z2, S3 or S4; correspondingly, Oi n Gyt is a union of 1, 2, 3 or 5 orbits.

The number of GY-orbits in which this yields is 28 or 7 according as G = E6 or D4; that is, we have 28 or 7 classes of p-elements in GYt. Therefore Lemma 3.3(i) implies that is the

Table 9. Classes of 2-elements in GY t, G = E6.

Class rep. in Gt no. of Gy-classes centralizer orders in Gy

vi 1 |F4(q)|

v2 1 qi5|Cs(q)|

v3 1 qi4|G2(q)|

v4 2qi5|As(q)|, 2qi5|2As(q)|

V5 1 qi8|Ai(q)||Ai(q)|

V6 1 qi9|Ai(q)|

V7 2qi0 |Ai(q)||Ai(q)|, 2qi0|Ai(q2)|

V8 1 qi5|Ai(q)|

V9 1 q8|G2(q)|

Vi0 1 qi3|Ai(q)|

vii 2 2qii|Ai(q)|, 2qii |Ai(q)|

Vi2 2 2q7Ai (q) |, 2q7|Ai(q)|

Vi3 1 q8

Vi4 2 2q9 (q - 1), 2q9(q + 1)

Vi5 5 24qi2, 8qi2, 4qi2,4qi2,3qi2

Vi6 2 2q6, 2q6

Vi7 2 2q4, 2q4

Table 10. Classes of 3-elements in GYt, G = D4.

Class rep. in Gt

vi 1 |G2(g)|

V2 1 q5|Ai(q)|

vs 1 q3|Ai(q)|

V4 3 6q4,3q4, 2q4

no. of Gy-classes

centralizer orders in Gy

complete set of outer unipotent elements in GYr. And as this holds for all q, we argue as in the proof of Lemma 3.3(ii) that this forces Q to be the complete set of outer unipotent elements in Gr.

Finally, we must verify that the information in the third column of Tables 9 and 10 is correct.

This procedure is just the usual Lang-Steinberg approach. Fix i ^ n and consider Ci = CG(ji). If the unipotent radical of this centralizer is Ud, then the group of fixed points has order qd.

Moreover, fixed points of 7 on Ri = Ci/Ru(Ci) are covered by actual fixed points. We must determine the possible actions of cj on Ri, where c G Ci/Ci° and cj is a representative of a conjugacy class in (Ci/Ci°)j.

A glance at Tables 7 and 8 shows that the only ambiguity occurs for G = E6 and i = 4, 5,

7 or 14, where Ri = D3.2, A1A1, A1A1.2 or T1.2 respectively. Lemmas 5.8, 5.9 and 5.13 show

that in the first, third and fourth of these cases, the extra involution induces the full group of outer automorphisms on the connected component of Ri. Consequently, two isomorphism types of fixed points are as indicated in Table 9. In the remaining case Ri is connected and the issue is whether or not 7 interchanges the A1 factors of Ri. However, if 7 interchanged these

factors, then it would also interchange the classes of 1-dimensional tori in their preimages. But we see from the argument in Lemma 5.8 that these tori are conjugate to the maximal tori of Y2 and y3j5, so this is impossible.

TABLE 11. Unipotent classes in Gt = A2t.

u v dim C C/Ru(C) e-function

Jl3 W(2) + У(2) 3 Sp2 1 ^ 1

J3 У(6) 1 2 3 ^ 1

Table 12. Unipotent classes in Gt = A3t.

u v dim C C/Ru(C) e-function

Jl4 W(2)2 10 Sp4 1 ^ 0

W (2) + У (2)2 6 Sp2 1 ^ 1

J22 W(4) 4 O2 2 ^ ш

J3,J1 У(6) + У(2) 2 1 3 ^ 1, 1 ^ 1

TABLE 13. Unipotent classes in Gt = A4t.

u v dim C C/Ru(C) e-function

Jl5 W(2)2 + У(2) 10 Sp4 1 ^ 1

J22,Ji W(4) + У(2) 6 Sp2 2 м- ш, 1 м- 1

J3, Jl2 У (6) + W (2) 6 Sp2 x 2 3 ^ 1, 1 ^ 0

У(6) + У(2)2 4 1 3 ^ 1, 1 ^ 1

J5 У(10) 2 2 5 ^ 1

TABLE 14. Unipotent classes in Gt = A5t.

u v dim C C/Ru(C) e-function

Jl6 W(2)3 21 Sp6 1 ^ 0

J22,Jl2 W (2)2 + У (2)2 15 Sp4 1 ^ 1

W(4) + W(2) 11 Sp2 x O2 2 ^ ш, 1 ^ 0

J3,Jl3 W(4) + У(2)2 9 SP2 2 м- ш, 1 м- 1

У (6) + W (2) + У (2) 7 Sp2 3 ^ 1, 1 ^ 1

J32 W(6) 7 Sp2 3 ^ 0

У(6)2 5 2 3 ^ 1

Jb, Jl У (10) + У (2) 3 2 5 ^ 1, 1 ^ 1

6. Tables

This section contains a number of tables illustrating our results. Tables 7-10, referred to in Theorems 1.4 and 1.5, cover the cases G = E6 and D4. Tables 7 and 8 give precise information on outer unipotent elements in Gt , while Tables 9 and 10 give corresponding information for the finite groups.

Tables 11-16 cover the cases G = Ai for 2 ^ l ^ 7; as elsewhere in this paper we write n = l +1, and take G to be simply connected so that G = SLn(K). In each of these tables, for each unipotent element v G Gt the second column gives the decomposition V I v as in (1), where V = V2n(K) is the corresponding symplectic module, and the first column gives the Jordan form of u = v2 on Vn(K). The third column then gives the dimension of C = CG(v), the fourth gives the reductive part C/Ru(C) of C, and the last gives the values of the e-function of [16] on the sizes of the Jordan blocks (see the preamble to Lemma 4.2).

TABLE 15. Unipotent classes in Gt = A6t.

u v dim C C/Ru(C) e-function

Jl7 W (2)3 + V (2) 21 Sp6 1 ^ 1

J22,Ji3 W (4) + W (2) + V (2) 13 Sp2 X Sp2 2 m- w, 1 m- 1

J3, Jl4 V (6) + W (2)2 15 Sp4 X 2 3 ^ 1, 1 ^ 0

V (6) + W (2) + V (2)2 11 Sp2 3 ^ 1, 1 ^ 1

J3, J22 V (6) + W (4) 9 Sp2 X 2 3 m- 1, 2 m- w

J32,Jl W (6) + V (2) 9 Sp2 3 ^ 0, 1 ^ 1

J5,Jl2 V (6)2 + V (2) 7 1 3 ^ 1, 1 ^ 1

V (10) + W (2) 7 Sp2 X 2 5 ^ 1, 1 ^ 0

V (10) + V (2)2 5 2 5 ^ 1, 1 ^ 1

J7 V (14) 32 7 ^ 1

TABLE 16. Unipotent classes in Gt = A7t.

u v dim C C/RU(C) e-function

Jl8 W(2)4 36 Sp8 1 ^ 0

J22,Jl4 W (2)3 + V (2)2 28 Sp6 1 ^ 1

W (4)+ W (2)2 22 Sp2 X O2 2 ^ w, 1 ^ 0

J24 W (4) + W (2)+ V (2)2 18 Sp2 X Sp2 2 m- w, 1 m- 1

W(4)2 16 O4 2 m- w

J3,Jl5 V (6) + W (2)2 + V (2) 16 Sp4 3 ^ 1, 1 ^ 1

J3,J22,Jl V(6) + W(4) + V(2) 12 Sp2 3 m- 1, 2 m- w, 1 m- 1

J32,Jl2 W(6) + W(2) 14 Sp2 X Sp2 3 ^ 0, 1 ^ 0

W (6)+ V (2)2 12 Sp2 3 ^ 0, 1 ^ 1

V (6)2 + W (2) 12 Sp2 X 2 3 ^ 1, 1 ^ 0

V (6)2 + V (2)2 10 1 3 ^ 1, 1 ^ 1

J42 W(8) 8 O2 4 ^ w

J5,Jl3 V (10)+ W (2) + V (2) 8 Sp2 X 2 5 ^ 1, 1 ^ 1

J5, J3 V (10)+ V (6) 6 2 5 ^ 1, 3 ^ 1

J7,Jl V (14)+ V (2) 4 2 7 ^ 1, 1 ^ 1

Appendix. Explicit U-centralizers

We have seen that the determination of the centralizers Co(vi) in Subsection 5.3 frequently uses knowledge of the subgroups CqV) and Cu(vi). The structure of these groups is given in Tables 5 and 6, but in one or two places more detailed information is required. We conclude by providing explicit expressions for the groups Cu (vi); Tables 5 and 6 summarize the results presented here.

We begin with a brief comment on the structure constants in G. Since all roots in the root system E are long, the only non-trivial Chevalley commutator relations are of the form [xa(ti), xp(t2)] = xa+p(Napt!t2), in which the structure constant Na,p is ±1. If G = E6 there is no ambiguity, since we are working in characteristic 2; however if G = D4 we must specify the choices made. We have taken Na p = 1 for the following ordered pairs of positive roots

(a,3):

(10o, 01 o), (oo0,01o), (001,01o), (110, 00o), (011, oo0), (o10, 10o),

(011,100), (011,000), (110, 001), (10o, 011), (001, 11o), (00o, 11o),

(011, 110), (111, 011), (110, o10), (o10, 111).

The structure constants were calculated using [3, Proposition 4.2.2]; as can be seen, for all a, ¡3 € E+ we have NaT,pT = Na,p. Since xa(t)T = xaT (t) for a € n, by taking commutators we see that the same is true for all a € E+.

TABLE A.1. Explicit U-centralizers for G = E6.

1,2 {x 00000 (ci)x 00100 (c2)x 01000 (сз)х 00010 (сз)х 10000 (c4,)x 00001 (C4)

x x 00100 (c5)x 01100 (ce)x 00110 (ce)x 11000 (c7)x 00011 (c7)x 01100 (ce)

X x 00110 (c8)x 01110 (cg)x 11100 (c10)x 00111 (c10)x 01110 (cn)x 11100 (c12)

x x 00111 (c12)x 11110 (c13)x 01111 (c13)x 01210 (c14)x 11110 (c15) 10011

X x 01111 (c15)x 11111 (c16)x 11210 (c17)x 01211 (c17)x 11111 (c18) 10111

X x 12210 (c19)x 01221 (c19)x 11211 (c20)x 12211 (c21)x 11221 (c21) X x 12221 (c22)x 12321 (c23)x 12321 (c24) : cj e K}

3 {x 10000 (c1)x 00001 (c1)x 00100 (c12)x 01000 (c2)x 00010 (c2)x 11000 (c3)

X x 00011 (c3)x 01110 (c32)x 01100 (c4)x 00110 (c4)x 11111 (c42)x 11100 (c5) 0 0 0 0 0 0

X x 00111 (c5)x 11110 (c6)x 01111 (c6)x 00000 (ti)x 00100 (t2)x 01100 (¿3) 000111

X x 00110 (t3)x 01110 (t4)x 11100 (t5)x 00111 (t5)x 01210 (t6)x 11110 (¿7) X x 01111 (t7)x 11210 (¿8)x 01211 (c1 + ¿8)x 11^11 (t6)x 12210 (¿9) X x 01221 (c1 c2 + c3 + t9)x 11211 (t10)x 12211 (tu)x 11221 (c1c3 + c4 + ¿11) X x 12221 (t12)x 12321 (t13)x 12321 (t14) : cj, tj e K}

4 {x 00100 (c1)x 00000 (c2)x 00100 (c3)x 01110 (c4)x 01110 (c5)x 01210 (c6)

X x 10000 (t1)x 00001 (t1)x 11000 (t2)x 00011 (t2)x 11100 (t3)x 00111 (t3) X x 11100 (t4)x 00111 (t4)x 11110 (t5)x 01111 (t5)x 11110 (t6)x 01111 (t6) X x 11111 (t7)x 11210 (t8)x 01211 (t8)x 11111 (t9)x 12210 (t10)x 01221 (¿10) X x 11211 (tn)x 12211 (¿12)x 11221 (¿12)x 12221 (¿13)x 12321 (¿14) X x 12321 (¿15) : cj,tj e K}

5 {x 00100 (c1)x 00000 (c2)x 01000 (c32)x 00010 (c32)x 00100 (c4)x 01100 (c5)

0 1 0 0 1 0

X x 00110 (c5)x 01100 (c6)x 00110 (c6)x 01110 (c7)x 01110 (c8)x 01210 (c9) 011011

X x 11000 (c3)x 00011 (c3)x 11100 (¿1)x 00111 (¿1)x 11100 (¿2)x 00111 (¿2)

X x 11110 (¿3)x 01111 (¿3)x 11110 (¿4)x 01111 (¿4)x 11111 (c5)x 11210 (¿5)

X x 01211 (¿5)x 11111 (c32c4 + c6)x 12210 (¿6)x 01221 (c3 + ¿6)x 11211 (¿7)

X x 12211 (¿8)x 11221 (¿8)x 12221 (¿9)x 12321 (¿10)x 12321 (¿11) : cj ,tj e K} 1 1 1 1 2

6 {x 00100 (c12)x 00000 (c2)x 01000 (c3)x 00010 (c3)x 00100 (c4)x 01100 (c5)

X x 00110 (c5)x 01100 (c6)x 00110 (c6)x 01110 (c12c32 + c3 + c72)x 01110 (cg)

X x 01210 (c9)x 10000 (c1)x 00001 (c1)x 11000 (c7)x 00011 (c7)x 11100 (¿1)

X x 00111 (¿1)x 11100 (¿2)x 00111 (c1 + ¿2)x 11110 (¿3)x 01111 (¿3)x 11110 (¿4)

X x 01111 (c7 + t4)x 11111 (c14c32 + c12c3 + c12c72 + c52 + c5)x 11210 (¿5) 1 0 1

X x 01211 (c1 + ¿1 + ¿5) X x 11111 (c12c32c4 + c12c3 + c12^-X x 12210 (¿6)x 01221 (c7 + ¿3 + ¿6)x 11211 (¿7)x 12211 (¿8) X x 11221 (c14c32 + c12c72 + c1c7 + c52 + c7¿1 + t8)x 12221 (¿9)x 12321 (¿10) X x 12321 (¿11) : cj,tj e K}

ROSS LAWTHER, MARTIN W. LIEBECK AND GARY M. SEITZ TABLE A.1. (Continued).

i Cu (vi)

7 {x 11110 (ci)x 01111 (c2)x 00000 (ti)x 00100 (ti)x 01100 (t2)x 00110 (¿3)

x x 0 1 110 (t4)x 11100 (t3)x 00111 (t2)x 01210 (¿5 )x 11110 (t6)x 01111 (¿7) x x 11210 (t6)x 01211 (t7)x 11111 (t5)x 12210 (t8)x 01221 (tg)x 11211 (t4) x x 12211 (tg)x 11221 (t8)x 12221 (t10)x 12321 (t10) x x 12321 (t11) : Cj ,tj e K, t42 = t52 + t5 }

8 {x 00000 (C12)x 11000 (c1)x 00011 (c1)x 00100 (t1 )x 00100 (t2)x 01100 (t3)

1 0 0 0 1 0

x x 00110 (t3)x 01100 (t4)x 00110 (t4)x 11100 (t5 )x 00111 (t5)x 01110 (C1) 011001

x x 11100 (t6)x 00111 (C1 + t6)x 11110 (t7)x 01111 (t7)x 01210 (C1t1 + t5 + t72)

1 1 0 0 1

x x 11110 (t8)x 01111 (t8)x 11111 (t2)x 11210 (tg)x 01211 (C1t3 + t7 + tg) 11011

x x 11111 (t10)x 12210 (tn)x 01221 (tn)x 11211 (t12)x 12211 (t13) x x 11221 (t4 + t13)x 12221 (C1 + t42 + t6)x 12321 (t14) x x 12321 (t15) : C1,tj e K}

9 {x 00100 (C1)x 00000 (C2)x 01000 (C2)x 00010 (C2)x 00100 (C3)x 01100 (C3)

010010

x x 00110 (C3)x 01100 (C4)x 00110 (C4)x 01110 (C4)x 01110 (C5)x 01210 (C6) 011011

x x 11100 (t1)x 00111 (t1)x 11110 (t1)x 01111 (t1)x 11111 (t2)x 11111 (t3) x x 11211 (t4)x 12211 (t5)x 11221 (t1 + t5)x 12221 (t6)x 12321 (t7) x x 12321 (t8) : Cj,tj e K}

10 {x 00100 (C1)x 00000 (C2)x 01000 (C2)x 00010 (C2)x 00100 (C3)x 01100 (C3)

x x 00110 (C3)x 01100 (C4)x 00110 (C4)x 01110 (C4)x 01110 (C5)x 01210 (C6) 011011

x x 11100 (t1)x 00111 (t1)x 11110 (t1)x 01111 (t1)x 11111 (t2)x 11210 (t2) 110001

x x 01211 (t2)x 11111 (t3)x 12210 (t3)x 01221 (t3)x 11211 (t4)x 12211 (t5) x x 11221 (t1 + t5)x 12221 (t6)x 12321 (t7)x 12321 (t8) : Cj ,tj e K}

11 {x 00100 (C1)x 01100 (C2)x 00110 (C2)x 01110 (C2)x 01110 (C3)x 01210 (C4)

011011

x x 10000 (C)x 00001 (C)x 11100 (t1)x 00111 (Z + t1)x 11110 (Z + t1)x 01111 (t1) 0 0 1 1 0 0

x x 11110 (t2)x 01111 (t2)x 11111 (t3)x 11210 (t4)x 01211 (t4)x 11111 (t2) x x 12210 (t12 + t1)x 01221 (t12 + t1)x 11211 (t4)x 12211 (t5) x x 11221 (Zt1 + t1 + t3 + t5)x 12221 (t6)x 12321 (t7) x x 12321 (t8): Z e F2, Cj,tj e K}

12 {x 01110 (C1)x 00000 (t1)x 10000 (t1)x 00001 (t1)x 00100 (t1 )x 00011 (Z)

x x 01100 (Z)x 00110 (Z)x 11100 (Z)x 01110 (t2)x 11110 (t2)x 01111 (t2) x x 01210 (t2)x 11110 (t3)x 01111 (t4)x 11111 (t4)x 11210 (t4)x 01211 (t3) x x 11111 (t5)x 11211 (t5)x 12211 (t32 + t42)x 11221 (t32 + t42) x x 12221 (t6)x 12321 (t6)x 12321 (t7) : Z e C1,tj e K}

13 {x 01110 (C1)x 00000 (t1)x 10000 (t1)x 00001 (t1)x 00100 (t1 )x 00011 (t2)

x x 01100 (t2)x 00110 (t2)x 11100 (t2)x 01110 (t3)x 11110 (t3)x 01110 (t3) x x 01111 (t1 + t3)x 01210 (t1 + t3)x 11110 (t4)x 01111 (t5)x 11011 (t12 + t5) x x 11210 (t5)x 01211 (t12 + t4)x 11111 (t6)x 11211 (t13 + t6) x x 12211 (t23 + t22 + t42 + t52)x 11221 (t42 + ¿52)x 12221 (t7) x x 12321 (t13 + t12t3 + t6 + t7)x 12321 (t8) : C1,tj e K, t12 = t22 + t2}

TABLE A.1. (Continued).

Cu (vi)

{x 00100 (ti 0

x x 00111 0

x x 11110 1

x x 11111 1

x x12221 1

x x12321 2

{x 00100 (t1 0

x x 11000 0

x x 00111 0

x x01210 1

x x01211 1

x x01221 1

x x11211 1

x x11221 1

x x12221 1

x x12321 2

{x 00100 (t1 0

x x 00110 0

x x 01110 0

x x 00111 1

x x 11110 1

x x11210 1

x x12210 1

x x01221 1

x x11221 1

x x12321 1

{x 00000 (0

x x 01100 1

x x 11110 0

x x 11111

x x12210 1

x x12211 1

x x12221 1

x x12321 2

x 00100 (t1)x 01100 (t2)x 00110 (t3)x 01110 (t12)x 11100 (t3)

1 1 1 0 0

t2)x 01110 (t12 + t1)x 11110 (t32)x 01111 (t22)x 01210 (t4) 1 0 0 1

t32)x 01111 (t22)x 11111 (t12 + t1)x 11210 (t5)x 01211 (t6) t12)x 12210 (t7)x 01221 (t8)x 11211 (t12 + t4)x 12211 (ts)x 11221 (t7) h~ t1

+ t22t32 + t4)x 12321 (tg) 4 + t1t4 + t22132 + tg) : tj e K}

x 10000 (Z)x 00001 (Z)x 00100 (t2)x 01100 (Z + t2 )x 00110 (Z + t2) 0 0 1 0 0

Z2)x 00011 (Z2)x 01100 (t3)x 00110 (t3)x 01110 (Z2 + t3)x 11100 (t4)

Z2 + t4)x 01110 (t1)x 11100 (t5)x 00111 (t5)x 11110 (t5)x 01111 (t5) 11100

t6)x 11110 (t7)x 01111 (Z + t7)x 11111 (Zt1 + Z + t7)x 11210 (t8)

)x12210 (tg)

+ tg)x 12211 (t10)

Z2t1 + Z2 + Zt2 + t4 + tg)

Z2t12 + Z2

Z3t1 + Z 2t2 + Zt4 + t5 + t10)

Z2t1t3 + Zt1 + Z 2t22 + Zt2t3 + Zt5 + t42 + t8)x 12321 (t11) t12): Z e F4, tj e K}

+ t1)x 10000 (t1)x 00001 (t1)x 00100 (t12)x 01100 (t12 +11) 0 0 1 0

t12 + t1)x 11000 (t1)x 00011 (t1)x 01100 (t12 + t1)x 00110 (t12 + t1)

t12)x 11100 (t2)x 00111 (t12 + t1 + t2)x 01110 (t12 + t1)x 11100 (t3) 0 0 1 1

t3)x 11110 (t3)x 01111 (t3)x 01210 (t14 + t12 + t3)

t13 + t12 + t2)x 01111 (t13 + t1 + t2)x 11111 (t13 + t12 + t3)

+ t32)x 01211 (t14 + t12 + t22 + t32)x 11111 (t3) t15 + t13 + t12t2 + t1t2 + t22 + t32)

t15 + t13 + t12t2 + t1t2 + t22 + t32)x 11211 (t32)x 12211 (t4)

t5)x 12221 (t16 + t15 + t14 + t13 + t12t3 + t1t3 + t32)

t6)x 12321 (t17 + t16 + t15 + t14t2 + t14t3 + t14 + t13t3

+ t12t22 + t1212 + t1t22 + t32 + t6) : tj e K}

01000 (Z)x 10000 (Z)x 00001 (Z)x 00100 (Z)x 00110 (Z)x 11000 (t1) 000100

t1)x 00110 (t1)x 01110 (t1)x 00111 (Z + t1)x 01110 (t12)x 11100 (t1) 1 0 0 1 1

t12)x 01111 (t12 + t1)x 01210 (t12)x 11110 (t2)

Zt1 + t12 + t2)x 11210 (t12)x 01211 (Zt1 + t12 + t2)x 11111 (t3)

t2 + t3)x 01221 (t2 + t3)x 11211 (t2 + t3)

Zt12 + t14 + t13 + t12 + t2 + t3)x 11221 (Zt12 + t14)

Zt13 + t13 + t22)x 12321 (t14 + t13 + t22)

t4): Z e F2, tj e K}

We now describe how we use a computer to obtain the groups Cu (vi). We begin with the element vi and write it as tsx, where x G U and s G NG(TG) (so that s corresponds to an element of the Weyl group); usually s = 1, but in some instances in G = E6 it is a

ROSS LAWTHER, MARTIN W. LIEBECK AND GARY M. SEITZ Table A.2. Explicit U-centralizers for G = D4.

i Cu (vi)

1,2 {x10 0 (C1)x00 0 (C1)x00 1 (C1)x01 0 (C2)x11 0 (C3)x01 0 (C3)x01 0 (C3)

x x111 (C4)x011 (C4)x111 (C4)x111 (C5)x12 1 (C6) : Cj e K}

3 {x01 0 (C1)x111 (t1)x011 (t1)x11 0 (t1)x111 (t2)x12 1 (t3) : C1,tj e K}

4 {x01 0 (C1)x10 0 (Z)x00 1 (Z)x00 1 (Z)x11 0 (t1)x01 0 (Z + t1)

x x01 0 (-Z + t1)x11 0 (t2)x011 (Z2 + t2)x11 0 (-Z2 + t2)x111 (-t1) x x12 1 (t3) : Z e F3, C1,tj e K}

5 {x100 (t1)x00 1 (t1)x00 0 (t1)x11 0 (t1)x01 1 (-t1)x11 1 (-t13)

0 00 0 001 110 010 0

x x011 (-t13 +112)x 111 (-t13 - t12 - t1)x111 (t13 - t12) x x12 1 (t2) : tj e K}

product of reflections in mutually orthogonal simple roots. We also take a 'generic' element g = Elpe^ xp(Kp) of U, where S = £+ n (E+)s, and the various Kp are regarded as indeterminates; we order the roots in S so that the roots outside Q precede those inside Q.

We form the commutator \g,vj\ = g-1 .x-1 .gTS .x, which we treat as a sequence of root elements corresponding to positive roots. This sequence is then passed through a simplifying program which reduces it to a canonical form; in this form the roots are taken in a fixed order compatible with height. If this canonical form is not the identity, we choose a root for which the coefficient is non-zero, and seek to make it zero by writing one of the Kp in the expression for g in terms of the remaining indeterminates. This gives a modified sequence for the commutator, which we pass through the simplifying program again, and the resulting canonical form will have fewer non-zero coefficients. We continue in this way until the canonical form has been reduced to the identity; at this point, the expression for g gives the form of an arbitrary element of Cu (vi).

The expressions obtained are given in Tables A.1 and A.2. Our notation in these tables is as follows. We write Cj and tj for arbitrary elements of K, with the exception that in two instances in Table A.1 a relation of the form tj2 = tk2 + tk holds. If it appears, Z stands for an element of a finite field F (usually F = Fp, but in one instance in E6 we have F = F4). If all Cj are set to be 0, the resulting expression gives a typical element of Cq!(vi).

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Ross Lawther DPMMS

University of Cambridge Cambridge CB3 0WB United Kingdom

ril10@cam.ac.uk

Martin W. Liebeck

Department of Mathematics

Imperial College

London

SW7 2AZ

United Kingdom

m.liebeck@imperial.ac.uk

Gary M. Seitz

Department of Mathematics University of Oregon Eugene, OR 97403 USA

seitz@uoregon.edu