990 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTYThus in this section, and indeed for the remainder of the chapter, we assume:
HYPOTHESIS 14.3.1. Either
(1) Hypothesis 13.3.1 holds with L/0 2 (L) ~ L3(2), and G is not Sps(2) or
U4(3); or
{2) Hypothesis 14.2.1 holds, and G is not h, h,^3 D4(2), the Tits group^2 F4(2)',
G2(2)' ~ U3(3), or M12·Observe that in case (1) of Hypothesis 14.3.l, parts (4) and (5) of 13.3.2 say
that Hypotheses 13.1.1, 12.2.1, and 12.2.3 are satisfied, and 13.3.1 is satisfied for
any KE Ct(G, T) with K/0 2 (K) ~ L 3 (2). Thus we may make use of appropriate
results from the previous chapters 12 and 13, including (in view of the exclusions
in 14.3.1.1) results depending on Hypotheses 13.5.1 and 13.7.1. Similarly the ex-
clusions in case (2) allow us to make use of results from the previous section 14.2.
As usual, we let Z := 01(Z(T)), Mv := NM(V), and Mv := Mv/CM(V).
NOTATION 14.3.2. In case (1) of 14.3.1, Lis the member of Cj(G, T) appearing
in Hypothesis 13.3.1, while in case (2), take L := 02 ( (0 2 (M n Mc)M) ). (Thus L
plays the role of the group "Y" in section 14.2.)
Observe:LEMMA 14.3.3. {1) L ::::J M.
{2) M = !M(LT).
{3) Na(T) ::::; M, and each HE H*(T, M) is a minimal parabolic described in
B.6.8, and in E.2.2 when H is nonsolvable.(4) V is a TI-set in M.
(5) Na(V) = Mv.
{6) If H::::; Na(U) for some 1 # U::::; V, then H n M = NH(V).
PROOF. Part (1) follows from 13.3.2.2 in case (1) of 14.3.1, and by construction
in case (2). Part (2) follows from 1.2.7.3 or 14.2.2.3, and (3) follows either from
Theorem 3.3.1 together with 3.3.2.4, or from parts (7) and (8) of 14.2.2. Further(5) follows from (2); and (4) follows by construction of M = Na(V) in case (2) of
Hypothesis 14.3.1, and from 12.2.2.3 in case (1). Finally as in the proof of 12.2.6,
(6) follows from (4) using 3.1.4.1. DWe typically distinguish the two cases of Hypothesis 14.3.1 by writing L / 02 (L) ~
L3(2) or L2(2)'.14.3.1. Preliminary results under Hypothesis 14.3.1.
LEMMA 14.3.4. If there exists KE Ct(G, T), then
{1) K/02(K) ~As or L3(2).
{2) K ::::1 KT and KE £,*(G, T).{3) Each VK E Irr+(K,R2(KT),T) is T-invariant, K, VK satisfies the FSU,
and VK is the natural module for K/02(K) ~As or L3(2).
(4) Case (1) of 14.3.1 holds, so that L/02(L) ~ L3(2).
PROOF. First case (1) of 14.3.1 must hold, since in case (2), Ct(G,T) = 0 by
Hypothesis 14.2.1.1. In particular, (4) holds. Further 14.3.1.1 excludes G ~ Sp 6 (2)
or U4(3), so K/0 2 ,z(K) is not A 6 by Theorem 13.8.1. Also we saw Hypothesis
13.5.1 holds, so (1)-(3) follow from 13.5.2. D