ii.i. THE SUBGROUPS Na(V1) FORT-INVARIANT SUBSPACES V 1 OF V 765(c) q = 4, K/Ooo(K) ~ L2(5), and 02(K) < 000 (K) centralizes Vi of rank at
least 4.
In case (b), K = 0
31(Na(Vi)) by A.3.18, so X 2 ::::; CK(L2/0 2 (L 2 )), contrary to the
structure of A1. In case (a), m(Vi) = 3n so m(Cv(L)) = n, and from the action
of L ~ Sp4(2n) or G2(2n) on Vin I.2.3.1.ii.a, R2 centralizes Vi, whereas this is
not the case for the parabolic L2 in K* ~ SL 3 (q) on Vr. Hence case (c) holds
so K/02(K) is not quasisimple and there is Y E S(G, T) contained in 02 ,F(K)
by 1.3.3; in particular Y is normalized by L 2 T. Next YT ::::; Ca(Vi)T ::::; Na(Vi),so as Y E 3( G, T), we may apply 1.3.4 to conclude that either Y :SJ Na(Vi), or
Y ::::; Ky E C(Na(Vi)) with Ky described in 1.3.4. Therefore either Ki :=::; Na(Y)
or Ki = Ky. However comparing the list of possibilities for Ky in 1.3.4 to the listof possiblities for Ki in this lemma, we find no overlap. Thus Ki :=::; Na(Y), so
LT= (Li, L2T)::::; Na(Y).
Then K::::; Na(Y) ::::; M = !M(LT), for our usual contradiction. This completes
the treatment of the case i = 2, and hence the proof of 11.1.2. DIn the remainder of the section, we obtain several further technical restrictions
on the normalizers of the subspaces Vi.LEMMA 11.1.3. L2 is the unique member ofC(Na(Vi)) which does not centralize
Vi.
PROOF. By 11.1.2, L2 E C(Na(Vi)). If there is L2 =f. KE C(Na(Vi)), then by
1.2.1.2, [K, L2] ::::; 02(L2) ::::; Ca(Vi), so as Vi E Irr +(L2, Vi), K centralizes Vi by
A.1.41. DLEMMA 11.1.4. Ca(Vs/Vi)::::; M :2:: Na(Vs) n Na(Li).
PROOF. If Lis SL3(q) then Vs= V, so Na(Vs) ::::; M = !M(LT). Hence we- L
may take L to be Sp4(q) or G2(q). Let ~ := V 2 1 and H := Na(Vs); note that
Vi is the intersection of the members of~' while Vs is their span. Then by 11.1.3,
NH(~) acts on
(L(U) : U E ~) = L,where we recall L(U) = NL(U)=. Therefore NH(~)::::; Na(L) = M. In particular
Ca(Vs/Vi) ::::; M. Further ~ is the set of subspaces Cv 3 (S) for S E Syb(Li), so
NH(Li) :SM. D
LEMMA 11.1.5. Either(1) Na(Ri) ::::; M, or
(2) V = VM, L is an SL3(q)-block or Sp4(4)-block, Cr(L) = 1 and Vi =
f!i(Z(Ri)).
PROOF. Assume Na(Ri) i. M. Then as M = !M(LT), there is no nontrivial
characteristic subgroup of Ri normal in LT. Therefore L, Ri is an MS-pair in the
language of Definition C.1.31. so L appears on the list of Theorem C.1.32. Therefore
Lis an Sp4(4)-block or an SL 3 (q)-block, since the remaining possibilities in C.l.34
explicitly exclude the case where Ri is the unipotent radical of the point stabilizer.
In particular V = V M.
Set Q := 02(LT) and Qi := VCr(V). If V = Q then fh(Z(Ri)) = Cv(Ri) =
Vi and the lemma holds, so we may assume V < Q. By C.1.13, q,(Q) :=::; Cr(L) and