1132 15. THE CASE .Cr(G, T) =^0
CAut(L)(Y+) = 1. Then by 15.3.29.2, V1 S Cs(Y+) S Cs(L), so LS Cc(Vi) SM
by 15.3.29.2, contrary to the choice of L. D
LEMMA 15.3.31. I* ~ Ss.PROOF. Assume otherwise; by 15.3.30 and 15.3.28, we may assume that case
(2) of 15.3.28 holds; that is L* ~ L3(2), Sp4(2)', or G2(2). As Vi= [V2, Y+] ~ E4 is
a Y+S-invariant line in UL, it follows from Theorem B.5.6 that MI, is the parabolic
stabilizing the line Vi in some W E Irr+ ( L *, UL), and hence for each such W when
[0 1 (Z(S)),L] is a sum of two isomorphic natural modules for L* ~ L3(2). By
15.3.29.1, (LS, S) is an MS-pair in the sense of Definition C.1.31, and by 15.3.27.3,
Lis not a A 6 -block. Therefore by C.1.32, either Lis a block of type A6 or G2(2), or
L ~ L 3 (2) and Lis described in C.1.34. In particular, L/0 2 (L) is simple in each
case, so that L/02(L) = L. Set Q := [0 2 (L),L]. As L ::::] I= LS, Q = [02(I), L].
In case (4) of C.1.34, m(0 1 (Z(S))) = 3, contrary to 15.3.8, and case (5) of
C.1.34 does not hold, as MI, stabilizes the line Vi. Thus only cases (1)-(3) of
C.l.34 can arise when L* ~ L3(2).
Next by 15.3.29.2, Vi S Cr(Y+) =: D. It will suffice to show that D = Cr(L):
for then L S Cc(V 1 ) S M by 15.3.29.2, contrary to the choice of L. Set J+ :=
I/Cr(L); it remains to show that D+ = 1.
Suppose that D centralizes Q. Then [D, Q] S CL(Q) S 02(L), so as L/Q is
quasisimple, [D,L] :::; Q. Thus [D,L] :::; CQ(Q) = Z(Q). Therefore 02 (D+) = 1
by Coprime Action. Further as q)(Z(Q)) = 1, q)(D+) = 1 (cf. the argument in
the proof of C.1.13); so as Y+ centralizes D, D+ = 1 from the structure of the the
covering of the L*-module Z(Q)/Cz(Q)(L) in I.2.3..
Therefore we may assume [D, Q] # 1. Thus Q i Y+· In case (3) of C.1.34,
Z(Q) is a natural module for L* and Q/Z(Q) is a sum of two modules dual to Z(Q).
In this case, and when Lis a block of type A 6 or G 2 (2), since MI, is the parabolic
stabilizing the line V2 in Z(Q), Q = [Q, Y+] S 02(Y+)· Therefore case (1) or (2)
of C.1.34 holds. Then Cr* (Y-t) = 1, so [D, L] :::; Q. Further the intersection of Y+
with each WE Irr +(L, Q) is a hyperplane W 0 of W, so as DQ ::::] DL and Q is
abelian, DQ centralizes (WJ') = W. Therefore DQ centralizes Q, a contradiction
completing the proof. D
Now L/02(L) is quasisimple by 15.3.26.2, L* ~As by 15.3.31, and the Schur
multiplier of As is a 2-group by I.1.3. Then as I= LS, 02(I) = Cr(UL) = Cs(UL)·
LEMMA 15.3.32. {1) UL is the 6-dimensional orthogonal module for I*.
{2) Cz 8 (L) =Zs n V1 =: Z1 is of order 2.(3) L = 031 (Cc(Z1)).
(4) X := 031 (Cc(Zs)) = 031 (K), where K is the maximal parabolic of L over
S n L determined by the end nodes of the diagram of L *.
(5) W :=(UL, U1,) =UL x U1 fort ET-S, and XS normalizes W.
{6) Let (XS)+ := XS/Cs(W) and P := 02(XS). Then p+ = Cs(UL)+ x
Cs(Ui)+.
PROOF. By 15.3.31, I* ~ Ss, and then by 15.3.28.3, MI, is the middle-nodeminimal parabolic. Therefore as UL is an FF-module for I*, B.5.1 says that either
UL/CuL (L) is the orthogonal module, or UL is the sum of a natural module and its
dual. Then as Vi= [Vi, Y+] is an S-invariant line of UL, the former case holds with
CuL(L) = 1, giving (1). Recall that Zs~ E4 by 15.3.8, and that Zi :=Zs n Vi is