588 3. DETERMINING THE CASES FOR LE .c:;ca, T)
(3) M+ ~ A 5 or A 7 , and [V, M+] is the natural module for M+.
(4) M+ ~ SL 3 (2n), Sp 4 (2n)', or G 2 (2n)', and [V, M+] is either the naturalmodule for M+ or the sum of two isomorphic natural modules for M+ ~ SL3(2n).
(5) M+ ~ A1, and [V, M+] is of rank 4.
(6) M+ ~ A5, and [V, M+] is of rank 6.
(1) M+ ~ L 4 (2) ·or L 5 (2), and the possiblities for [V, M+] are listed in Theorem
B.5.1.1.
PROOF. By 3.3.7.2, Vis an FF-module for M+T, and by 3.3.7.3, 02(M+f') =
- Hence the action of J := J(M+T, V) on [V, J] is described in Theorem B.5.6.
In case (2) of Theorem 3.3.1, M+T is a minimal parabolic, and using 3.3.7.1,
M+ is noncyclic, so conclusion (1) of the lemma holds by B.6.9. Thus we may
assume case (1) holds. Therefore F(M+f') = M+ =Lor LD fort ET-NT(L).
Therefore as 1 -I J :::;! M+f', M+ = F(J). Further if L < M+, then L ~ L2(2n),
Sz(2n), L 2 (p) or J 1 by 1.2.1.3. Therefore conclusion (1) of the lemma holds by
B.5.6.
Thus we may assume that L = M+, so that L = F(J) = F(M+T) is qua-
sisimple. Hence the action of Lon Vis described in Theorem B.5.1. The conclusions
of the lemma inciude cases (ii), (iii), and (iv) of B.5.1.1 in which [V, L] is reducible,
so we may assume [V, L] is irreducible. Hence by B.5.1 the possibilities for the ac-
tion of LT on [V, L] are listed in Theorem B.4.2, and again our conclusions contain
all those cases. DLEMMA 3.3.9. cM+(Z) = cM+(Zn [V,M+D·
PROOF. Since Z =(Zn [V, M+])Cz(M+) by 3.3.7.4, the lemma follows. D
We now begin to eliminate cases from 3.3.8:LEMMA 3.3.10. (1) If H E 1i(T) and T is contained in a unique maximal
subgroup of H, then 02 ( (H, D)) -I 1.
(2) M+ is not L2(2n), eliminating case (2) of 3.3.8 and the A5-subcase of case
(3) of 3.3.8.
(3) If case (1) of 3.3.8 holds, then Li~ L3(2).
(4) Case (1) of the hypothesis of Theorem 3.3.1 holds.
PROOF. Part (1) follows from Theorem 3.1.1, with TD, Tin the roles of "M 0 ,
R". In particular if T lies in a unique maximal subgroup of M+T, then (1) contra-
dicts 3.3.6.b. Parts (2) and (3) follow from this observation. Finally, as we observed
earlier, if case (2) of the hypothesis of Theorem 3.3.1 holds, then conclusion (1) of
3.3.8 holds with Li~ Z3. Thus (3) implies (4). DREMARK 3.3.11. By 3.3.10.4, case (1) of Notation 3.3.4 holds. Therefore M+ =
(LT), where L E .C*(G, T) with L/0 2 (L) quasisimple. Thus L has this meaning
from now on.
LEMMA 3.3.12. Suppose YE .C(L,T) and 02 (H) # 1 where H := (Y,TD).
Then
(1) Y::; KE C(H).
(2) K :::;! H.