Adjoint Module

In this section we will briefly introduce the concept of adjoint modules. For a more detailed discussion see [8, Section 5.4.1, p.293].

Definition 8.4.1. Let ρ be a representation of some group and let V be the module ofρ. ThenV˚ is the_{dual module} acted on by_ρ´T_.

Let G“GL˘tpqqand let gPG. LetV be the natural module of Gover

Fqu and letV˚ be the dual module of V. Define a representation ρ :GÑ

V bV˚ _{by gρ}

“gbg´T_{. Let M be the}

FqGLtpqq-module MtˆtpFqq or let M be theFqGUtpqq-moduleM “ tAPMtˆtpFq2q |AT “Aσucorresponding

toGρ. Here σ sends the entriesaij of A toaq_ij.

Definition 8.4.2. Let M be as above. Let U be the submodule of M consisting of all matrices of trace 0 and let U1 be the submodule of _M consisting of all scalar matrices. Then theadjoint moduleW is

W “U{pU XU1q.

Lemma 8.4.3 ([8, Lemma 5.4.10, p.294]). Let G“ GL˘_tpqq and let W be as in Definition 8.4.2. If p | t then W has dimension t2´2. Otherwise W has dimension t2 ´1. Furthermore, W is absolutely irreducible as an

FqSL˘tpqq-module.

By [8, p.294] we can define a quadratic form Qon M by

QpAq “ ÿ

1ďiăjďn

Throughout the reminder of this section let B be the form matrix of the polar formβ of Q. Let E be the subset of M containing the matrices with all diagonal entries equal to 0 and letD be the set of diagonal matrices of M.

Lemma 8.4.4 ([8, Lemma 5.4.11, p.295]).

Let G “ GL˘t pqq and let M, U, U1, W, E, D and β be as defined above. Then:

(i) W –E K pDXUq{U1_;

(ii) E is a non-degenerate space of plus-type if G “ GLtpqq and a non- degenerate space of type p´1qp

t

2q _{if G“}GU_tpqq;

(iii) If p|tthen β is degenerate.

Hence we have to determine the type of orthogonal form of the non- degenerate spacepDXUq{U1_{. We will only have to consider the case when} p“2 andt“4.

Lemma 8.4.5. Let G“ GL˘tpqq and let M, U, U1 and D be as above. If p“2 and t“4, then pDXUq{U1 _{is a non-degenerate space of minus-type}

in any odd extension of_F2 and an orthogonal space of plus-type in any even

extension of_F2.

Proof. By the proof of [8, Lemma 5.4.11(iv), p.295], DXU “ xd1, d2, d3y,

where dj “Ej,j´E4,4 for all 1 ď j ď 3 and Ei,j “ palkq is a matrix with alk “1 ifl“i,k“j and alk “0 otherwise. Furthermore, the form matrix with respect to this basistd1, d2, d3uis

B“

´_{0 1 1}

1 0 1 1 1 0

¯

and Qpdjq “1 for allj. Let U1 _{“ xdy, where d}

“diagp1,1,1,1q “d1`d2 `d3. It is clear that

pDXUq{U1 _{“ xd}

1`U1, d2`U1y. We now have to find the quadratic form

Q1 with polar form _β1 on _pDXUq{U1. We have Qpdj`αdq “Qpdjq `Qpαdq `βpdj, αdq “1`α2Qpdq `αpβpdj, d1q `βpdj, d2q `βpdj, d3qq “1`6α2`2α “1 for allαPF2i. Ifk ‰j thenβ1<sub>pdj</sub> `U1_{, dk} `U1 q “βpdj, dkq “ 1 by Lemma 3.3.3. Hence, the matrix ofQ1 <sub>is</sub>`_{1 1} 0 1 ˘

which is of minus-type if and only if x2`x`1 is irreducible inF2i by [8, Prop 1.5.42(iii), p.24].

It is clear that x2`x`1 is irreducible in F2 and from [28, Cor 3.47,

p.100] it follows thatx2`x`1 is irreducible in any odd extension ofF2 and

reducible otherwise.

Corollary 8.4.6. LetG“SL˘₄p2iq and letU,U1_{,W,E andDbe as above.}

Then the adjoint module W of Gpreserves an orthogonal form of plus-type if and only if iis even and an orthogonal form of minus-type otherwise. Proof. By Lemma 8.4.4, W – E K pDXUq{U1 and _E is a space of type k1 “ `. Letk2 be the type of the spacepDXUq{U1. By [8, Prop 1.5.42(iv),

p.24],W has typek1k2. The result follows from Lemma 8.4.5.

Lemma 8.4.7 ([8, Lemma 5.4.13, p.297]). Let ρ be the adjoint representa- tion of SL˘

tpqq and let d generate the diagonal automorphisms of SL˘t pqq. ThendρPSOnpq, Bq. Furthermore, dρPΩnpq, Bq if and only if t is odd or q is even.

Lemma 8.4.8([8, Lemma 5.4.14, p.297]). Suppose thatG“SL˘t pqq has an adjoint representation ρ of dimension n. Then γ P OutpL˘tpqqq is induced by an element g P GO_npq, Bq. If `₂t˘ is even or if qn is odd then g P Ω

npq, Bq or ´g P Ωnpq, Bq. If `_t

2

˘

and q are odd and n is even then g P GO

npq, BqzSOnpq, Bq. If `_t

2

˘

is odd and q is even then we can show that gPSO_npq, BqzΩ

npq, Bq.