The Structure of the Unipotent Radicals

Preliminary Lemmas

7.1 The Structure of the Unipotent Radicals

Let c j be the flag

Q < V h < V h < - . . < V js.

Set B = Ng(c[), Pj = Ng{cj), and Pq = G. Then B is a Borel group of G and {P j; J C / } is a set of parabolics of G over B . Let P j = UjLj b e th e Levi decompo­

sition of P j, w ith Uj and L j being the unipotent radical and Levi factor, respectively, as in section 3.1. As usual, we w rite Pj for Pyy. It is well known th a t

Uj

=

c a <yj)

n

C oiyj-iv,)

n

c a ( v / v +).

L e m m a 7 .1 .1 . Fix j e I .

(1) Z (U j) = C(/j (Vj1~) = Gjj.(Vj) where V- = (eJ+1, . . . , I n particular, as an additive group, Z(U j) is isomorphic to M jj(F q) and U jfZ (U j) is isomorphic to M j,n-2j(F qI). (Uj = Z j when n = 2 j ) Z(U j) is an FqLj-module while Uj/Z(Uj) is an ¥ qiLj-module, both induced from the conjugation o f L j on Uj.

(2) L j = N c(V j) (1 Ng(VJ) D N g (V " ) where V" = (en_J+1, . . . , e*}. In particular Lj

=

Lj x LJ where L+

=

CLj(Vj) S G+i and L j = C Li{VJ) = Gn~2j.

Moreover, L j acts trivially on Z(Uj); Uj/Z(JJf) is the tensor module fo r Lj described in the beginning o f section 6.1.

Proof. Pick u G Uj. As a m atrix, u can be written as:

(

^{I ■}13J ^{0 0}

^

u = A In—2j,n—2j ⁰

{ °

^{B h o )}

where I j j is th e identity j x j m atrix, A is a (n — 2j) x j m atrix, etc. Certainly we m ust have M = u M u Te. Indeed all the assertions in this lem m a can easily be deduced from direct calculation, and we omit the proof.

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Similarly for J C / represented as in equation (7.1), it is well known th a t U j is the subgroup of G centralizing the successive quotients of the following series:

0 <

Vh

<

vh c - . V j . a v £ < v £ < V £ < v .

L e m m a 7 .1 .2 . (1) For 1 ^ i ^ s, U j ^ j () is a norm al subgroup o f P j as it is the unipotent radical of a subparabolic o f P j. Consequently the following is a chain of normal p-subgroups o f P j:

Uj = O o * )

> =

Uj. > Z,. > 1. (7.2)

(2) For 1 ^ i < s,

V(ji,3i+1) = Uj&rt/UjGzji+i) =

as an abelian group. V (ji,ji+1) is an F92P j ) -module induced by conjuga­

tion. P j / U j ^ j i ) is isomorphic to

x P j * x G+Ui+1~ji) (7.3)

where J0 = {j — j i+i | ji+i < j € J } and J x = J ( < ji). acts trivially on V (ji, ji+i) by conjugation. When regarded as a module fo r P j ^ 1 x G+^ i+l~3i\

V (ji, ji+i) is a tensor module as in Example 6.1.1. Consequently P j is the semi-direct product o f Ujfeji+l) by

j-(n—2ji+i) p+ii+l -K/b x OiUffi}-

(3) As an abelian group, Z(Ujt ) = M jsj t {¥q) while U j,/Z (U jt ) = Mjttn-2j,QFq2)- Z(Uj.) is an ¥ qP j/U jt -module while U jj Z ( U j t ) is an ¥ q2P j/U jt -module, both induced by conjugation. Moreover, P j/U jt is isomorphic to

P+J' x CP-25’ (7.4)

where Ji

=

J (< j s). Gn~2^ acts trivially on Z{Ujs). Ujs/Z{Ujs) is a tensor module f o r Pj/Uj3as in Example 6.1.1.

Again these assertions are either well known o r can easily be deduced from direct calculation.

We stu d y a “tw isted” variation of a tensor module.

Let G

=

GU2m(<l) for some m E N. Let P

=

Pm. So L — L m

=

G L m (q2). Let V = Um = Op(P). From section 7.1, we know V is an F9L-module via conjugation.

So by Lemma 6.1.2 and Rem ark 6.1.3, Irr(V ) is L-isomorphic to th e dual module V*. O n the o th er hand, the unipotent radical V~ of th e opposite parabolic P~ is known to be isomorphic to th e dual m odule for V . Therefore, in this situation we may as well identify V ~ with Irr(V').

Let G = G L2m(F), where F is th e algebraic closure of F?. Fixing a basis for the n atu ral module on which G acts, we m ay view G as a m atrix group. Define a : G -> G by (aij) M - I (a?-)- r M where X T is th e transpose of X and M = (m ,j) w ith

rriij —

1, i f i + j = 2m + 1;

0, otherwise.

So cr is an extended Frobenius endom orphism of G and we may assume Ga = G.

We can choose a cr-stable maximal parabolic subgroup P w ith Pff = P . T hen V = Op(P) = MmiTn(Fq) is cr-stable and Va = V . Also V has a cr-stable Levi complement L = G L m(Fq) x G L m{f 9) w ith L ff = L. As V ~ is characteristic in P ~ and P ~ is cr-stable, V ~ is cr-stable and (V'")<r = V ~ .

By th e set-up in Example 3.1.1 an d th e choice of cr, we have L = L i x L2 with L i — L2 — G Lm (¥ qa)

w ith cr(Li) = L z-i, i = 1,2. W ritten as a (2m ) x (2m )-m atrix, a typical element

gi E L i has the form

A 0

0 I m , m j

where A is a non-singular m x m -m atrix. A typical element <72 6 has the form Im,m 0

0 B t

where B is a non-singular m x m -m atrix. Consequently a typical element g E L has the form

A 0

°

V ~ can b e either upper triangular or lower triangular. We assume th e former. Then a typical element x E V ~ has th e form

Im,m C

0 I m , m ,

w ith C E MmiTn(Fg). So each x is uniquely determ ined by an m x m -m atrix. Clearly gxg~l has th e form

W A C S" 1

0 I m , m

Moreover, if x E V ~ corresponds to C = (c#) as above, then a (x) has th e form Im,m {di,j)

0 f m ,m

As both. V ~ and Irr(V’) are dual to V as FL-modules, we m ay identify them with each other. So th e results apply when we study the action of L on V ~ , as L = L\ x L2 and V is a tensor module for L. In particular, Lemma 6.1.7 says L has 1 -+- m orbits on V~ and describes the orbit stabilizers.

On the other hand, we may identify V ~ as Mm>m(F), as we do in Exam ple 6.1.1.

As th e action of L on V~ preserves the rank of matrices, we conclude th a t th e Z-orbits on V ~ are determ ined by the ranks.

Next we define a set of representatives of V ~ / L as follows. Fix 0

^

e

6 f ?

with e9 + e = 0. For 1 < r ^ m, let x r = (a^-) € V ~ be defined as:

{

^e, if j — i = m — r;

0, otherwise.

By definition x r has rank r as a m atrix. So {0, x r\1 ^ r ^ m} is indeed a set of representatives of V~/L. Moreover, it is easy to check th a t xT is <x-stable for all r.

Assume V ~ = V\ <g> V2 as in Example 6.1.1. So VI is the n atu ral m odule for L x while V2 is the dual of the natural module for L 2. Fix 1 ^ r ^ m and let H = N i ( x r).

So by Lemma 6.1.7.2,

C = C 1x C2 ^ H ^ l x H 2

such th a t Hi = N ’i.(R i) for some co-dimension r subspace Ri of Vi, an d Ci — C Zi(Vi/Ri). Moreover,

Hi = U i * (K i x K<) where

U, = C e ,{ R ,) Si M r m -r(F)>

K i = GLm_r (F) is the stabilizer in Cr o f a complement ilj to Ri in Vlt K i = Ca (Bi) E£ G L r(W),

Ci = UiKi, and

H = (C i x C2) x D,

where D = G L r (F) is a full diagonal subgroup o f K[ x K'2.

Now as x r is cr-stable, and cr(Lt) = X3_j for i — 1,2, it follows th a t a’(Hi) = Hz cr(Ki) = , and cr(K^) = so cr(Ci) = C3-1, and as xr is cr-stable, cr(D) = D.

We deduce th a t

C = Ca = U x K where

U = (£ W 2)ct “ Mr.m_r (F ,2),

and

H = lVL(a;r ) = = C£>, where

D — 0 ^ = GUr ( F # ) . By construction,

C = Cl(V0/ Ro) ^ ^ WL(ilo)

where Rq is a. co-dimension r subspace of th e natu ral module Vq for L. We have shown

L e m m a 7 .1 .3 . Let 1 ^ r ^ m .

(1) There is a unique L-orbit on the elements o f V ~ of rank r, and {0, x r ;1 ^ r ^ m } is a set of representatives of L-orbits on V ~ .

(2) N i ( x r) is isomorphic to

( (Mr,r ( f ) x GZrm_r (F)) x (Mr,r (F) x GLm_r ( f ) ) ) x GLr ( f ) .

(3) cr(xr) = x r, and there is a co-dimension r subspace R o f the natural module V m f o r L such that

N L(xr) = U » ( K x D ) stabilizes R,

U = CL{R)

n

C L(V m/ R ) 3 Mr)m_r (Fq2)f K = N l {R) D Cl (R') = GLm_r (F ,2), where V m = R © R ', and

d = N L(xT)

n

ML(R')

n

c l (R) = GUr (¥q).

By Lemma 7.1.3, cr acts on each. L -orbit on V ~ . Therefore, studying th e L-orbits on V ~ is equivalent to studying how th e fixed points by a in each orbits breaks into orbits of V ~ under L-action.

L e m m a 7 .1 .4 . For 1 ^ r ^ m , x% is the unique L-orbit on the set o f elements o f rank r o f V ~ . Consequently L has 1 + m orbits on V ~ with {0, x r; 1 ^ r ^ m } being a set o f representatives o f V ~ / L .

The proof is an easy application of th e following well known lemmas.

L e m m a 7 .1 .5 . Let G be a group acting transitively on a set X , x E X , H = Gx and K ^ H . Then Nq{K) is transitive on F i x ( K) i f and only i f K G D H = K H.

Proof. This is (5.21) on page 19 in [As].

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L e m m a 7 .1 .6 . (Lang-Steinberg Theorem.)

I f H is a connected algebraic group over Fq and a is an endomorphism o f H , then the map g i-> cr{g)g~1 from H to itself is surjective.

Proof This is Steinberg’s generalization of Lang’s Theorem. See th e discussion on page 32 in [Ca]. For the proof, see [S].

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Proof o f Lemma 7.1-4- Let X = x L where x = x r. It suffices to show th a t L is transitive on X a.

Set G = (L, a) such th at ga = a(g) for g E L. T hen G acts transitively on X with Gx = L xT where T = (cr). Then N q ( T) = L T , F ix ( T ) = X a. By Lemma 7.1.5, L T is transitive on X a if and only if fl G x = T ^ x if an d only if fl L XT = T ^ x if and only if <x£cr- 1

n

L x = cr^cr-1 . Here cr^cr- 1 = {g~^cjga~x \ g E L}.

C ertainly L is a connected algebraic group. By Lemma 7.1.3, Lx is the semi-direct product of connected algebraic groups an d hence connected. So by Lemma 7.1.6, a La ~ l = L and o^cr- 1 = Lx. Therefore, th e final equality indeed holds. Hence the lemma is proved.

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Recall the ran k of x E V ~ as a m atrix is L-invariant. For each 1 ^ r ^ m , we set Irr(V, r) to be th e set of r 6 Irr(V ) identified w ith th e rank r elements in V ~ . we let r r ^G Irr(V, r) be identified w ith x r .