1066 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L:r(G, T) IS EMPTYq(H*, W) :::; 2, and that module is not a strong FF-module. By B.4.6.1, the Weyl
module is the unique indecomposable extension of that irreducible by a module
centralized by K*. By 14.7.30, we may apply parts (6) and (4) of F.9.18, so if
the lemma does not hold there are exactly two noncentral chief factors W1 andW 2 for H* on U, and each is of dimension 6. Indeed as in the proof of F.9.18.6,
m(U /Cu-(U 1 ) = 2m(U;) = 6 and u; is an FF*-o:ffender on both W 1 and W 2. Then
14.5.18.2 supplies a contradiction, as u; does not act as a group of transvections
on any subspace of corank 3 in U. · D. In view of 14.7.54, we now appeal to B.4.6 and [Asc87] for the structure of
U, and we use the terminology in [Asc87], such as "doubly singular line". As
V = [V, L 1 ] is T-invariant, we have:LEMMA 14.7.55. (1) V is a doubly singular line of U.
(2) fT2 is a singular point of U.
(3) The set V(Vi, Vi) of doubly singular lines in U through fT2 is of order 3,
and generates a subspace U(Vi, Vi) of rank 3. ·
(4) Cx.(U(V1, Vi)) =: B = B(Vi, Vi) ~ E4, [U,b] E V(Vi, Vi) for each
1 =f. b* E B*, and
w := W(Vi, Vi) := (Cu-(b*) : 1 =!= b* EB*) = -v;J_
is a hyperplane of U. If H* ~ G2(2), then CH* (U(Vi, V2)) =:A* = A*(Vi, Vi) ~
Es, and U(Vi, Vi)Cu-(H) = Cu-(a*) = Cu-(B*) = [U, A*] for each a* EA* - B*.
(5) If U is an FF-module for H* then H* ~ G 2 (2) and A*(V 1 , V 2 )H is the setof FF-offenders in H.
(6) LetY := 02 (CH(Vi)). ThenYT/02(YT) ~ 83, U(Vi, Vi)= [U(V1, Vi), Y],
and Y is transitive on V(Vi, Vi).(7) The geometry Q(U) of singular points and doubly singular lines in U is the
generalized hexagon for G 2 (2). In particular, there is no cycle of length 4 in the
collinearity graph of Q(U).
(BJ {[U,bJ: b EB}= {[w,AJ: w E w}.
In the remainder of this subsection, we adopt the notation in 14.7.55.
LEMMA 14.7.56. Let V(Vi) be the set of preimages in U of members ofV(V 1 , Vi),
and U(Vi) the preimage of U(V1, Vi). Then
(1) V(Vi) =VY is the set of G-conjugates of V containing Vi.
(2) Y centralizes L2/02(L2), and G 2 = L 2 YT acts on U(Vi), with L 2 fixingV(Vi) pointwise and G2/Ca(U(Vi)) the stabilizer in GL(U(Vi)) of Vi.
PROOF. By parts (1) and (6) of 14.7.55, vY = V(Vi). Then U(Vi) = (VY),
so as [V,L2] = Vi, while [L2, Y] :::; CL 2 (Vi) = 02(L2) by 14.7.4.2, we have
[U(Vi), L2] =Vi, and hence L 2 fixes VY pointwise. Further as H = G1, Ca(Vi) =
CH(Vi) = YCy(V2), so since L 2 T induces GL(Vi) on Vi, G2 = L2YT. Then
V^02 =VY, so as Lis transitive on the hyperplanes of V, (1) follows from A.1.7.1.
Finally Po := Auta 2 (U(Vi)) :::; NaL(U(V 2 ))(Vi) =: P with P = Po02(P) and
1 =/= 02(Auty(U(Vi))) :::; 02(Po), so P = P 0 as Pis irreducible on 02 (P). This
completes the proof of (2). D