PROOF. By Theorem 7.0.1, Vis an FF-module for AutaL(V)(L). By construc-
tion in the FSU, V = (V!) for some Vo E Irr +(L, R 2 (LT), T), so Vis T-invariant.
If V > V 0 , then Vis described in case (3) of Theorem 3.2.5. However in that case
by Theorem B.5.1, Vis not an FF-module for AutaL(V)(L). Therefore V =Vo, so
as Vis an FF-module, (2) follows since one of cases (2), (3), or (4) of 3.2.8 musthold. Then as V is T-invariant, T is trivial on the Dynkin diagram of L/02(L),
completing the proof of (1). DLEMMA 11.0.3. (1) VM E R2(M).
(2) VM is a homogeneous F2L-module.
(3) EitherCv(L) = CvM(L) = 1; or L ~ Sp4(q) orG2(q), V = VM, m(Cv(L))
::; n, and L = [L, J(T)].(4) V is a TI-set under M.
(5) If Lis Sp4(q) or G2(q) then H n M::; NM(V) for each HE H*(T, M).
PROOF. Part (1) is 3.2.2.2; part (2) follows from 3.2.2.3; and as n > 1, part ( 4)
is a consequence of 3.2.7. By 3.2.2.4, CvM(L) = (Cv(L)M). If L ~ SL3(q), then as
n > 1 we have H^1 (L, V/Cv(L)) = 0 by I.1.6, so Cv(L) = 1. Hence CvM(L) = 1,
so that (3) holds in this case. If Cv(L) "I-1, then L = [L, J(T)] by 3.2.2.6, and
V = VM by Theorem 3.2.5, since now neither cases (2) nor (3) of that result hold.Further by I.1.6, m(Cv(L)) ::; m(H^1 (L, V/Cv(L)) = n, completing the proof of
(3)..
Finally assume the hypotheses of (5), and suppose HE H*(T, M) with HnM 1.
. NM(V). In particular V < VM as VM '.SJ M. As Vis a TI-set under M by (4), while
ZnV-/-1, [Z, HnM] -/- 1 and hence [Z, HJ -/-l. Thus J(T) 1. Cr(V) by 3.1.8.3, and
so L = [L, J(T)]. So setting M := M/CM(VM ), by B.2.7 there is A E P(M, VM)
with L = [L,A]. Then by Theorem B.5.6, F(J(M,VM)) = L*, and then
Theorem B.5.1 supplies a contradiction to V < VM. D
LEMMA 11.0.4. L = QP' (M) for each prime p such that
(1) p divides q^2 - 1, if L is Sp4(q) or G2(q); or
(2) p divides q - 1 and p > 3, if L is SL3(q).
Moreover if L ~ SL3(q) with n even, then L contains each element of M of order
3.PROOF. The primes pare chosen so that mp(L) = 2; hence the lemma follows
from A.3.18, using A.3.19 for the final assertion. D11.1. The subgroups NG(Vi) for T-invariant subspaces Vi of V
By 11.0.2.2, V is the natural F qL-module; thus the two classes of maximal
parabolics of L preserve F q-subspaces of dimension 1 and 2. We will use our
structure theory of QTKE-groups to restrict the normalizers of these subspaces.
The results in this section roughly have the effect of forcing these normalizers to
resemble those in the shadows mentioned earlier.For i = 1, 2, 3, let Vi denote the set of U::; V such that Cv(L) ::; U and t) is
an i-dimensional FqT'-subspace of V for some T' E Syl 2 (M). Further for i = 1, 2,