i3.8. FINISHING THE TREATMENT OF A 6 959and H* ~ 87, B.4.2 and B.4.5 say that UH is either a natural module or the sum
of a 4-dimensional module and its dual. As V 3 is of rank 2 and T-invariant, with
V3 = [V3,Li,+] :S: CuH(Lo), we conclude that UH is natural, and Li,+= ((5,6, 7)).
Recall v; :::: 02(LiT*) = 02(Li,+)·' As v; is quadratic on UH by 13.8.4.6, it
follows that m(V;) = 1, so u H induces transvections on u"Y by 13.8.18.3. But then
v;y induces transvections on UH, whereas v;y :::: 02(LiT*), which contains no
transvections.
Thus LiT is the stabilizer of a partition of type 23 , 1. In particular m 3 (Li) =.1,
so L/02(L) ~ A6 as Li = e(H n M) by 13.7.3.9. As u; E Q(H*' UH), B.4.2 and
B.4.5 say that UH is either of dimension 4 or 6, or else the sum 4 + 4' of 4 and
its dual 4'. But Li stabilizes the T*-invariant line V 3 :S: UH, so as LiT is the
stabilizer of a partition of type 23 , 1, dim(U H) # 4 or 8, and hence dim(U H) = 6. If
[W,K] = 1, then [UH,K] ~UH is the natural module for K*, so as L/0 2 (L) ~ A 6 ,
13.7.6.3 supplies a contradiction. Thus [W, K] # 1, and so we may apply 13.8.22.
As H* has no strong FF-modules, we conclude from 13.8.22 that u; induces a
group oftransvections on UH or W. Therefore m(U;) = 1, and we saw this suffices
to complete the proof. DLEMMA 13.8.25. If K/0 2 (K) ~ Ln(2) for 3 :S: n :S: 5, then n = 4 and
{ 1) L / 02 ( L) ~ A6, and UH is a 4-dimensional natural module for H* ~ L4 (2).
{2)H=Gi.
PROOF. Assume otherwise. If case (1) of 13.8.8 holds, then conclusion (1) holds
by 13.8.5. In particular n = 4, and we will see below that this implies conclusion
(2); so we may assume that case (2) of 13.8.8 holds.
Then u; E Q(H,UH)· Let TK: := T nK*. As Li :S: K by 13.8.21.1,
LiTK: is a T-invariant parabolic of K. Indeed LiTK: is a minimal parabolic when
L/0 2 (L) ~ A5, since ILil3 = 3 in that case, whereas LiT /02(L]_T) ~ 83 x 83
when L/02(L) ~ A5.
If Ll_TK: is a minimal parabolic, then as LiTK: is T-invariant, either T = TK:,
or n = 4 and L]_TK: is the middle-node parabolic of K*. This allows us to eliminate
the case n = 3: For if n = 3, then m 3 (K) = 1, so L]_TK: is a minimal parabolic and
hence T* = TK- contrary to 13.8.21.3.
Further if n = 5, then L/0 2 (L) ~ A5: For otherwise we have seen that LiTK: is
a minimal parabolic and T* = TK- Therefore LiT :S: Hi :S: H with Hi/02(Hi) ~
83 x 83. But now Hi::;: M by 13.8.13, so Li= ()(HnM) is of 3-rank 2 by 13.7.3.9,contradicting L / 02 ( L) ~ A6.
In the next few paragraphs, we assume L/02(L) ~·A6 and derive a contradic-
tion. Here the arguments above have reduced us to the case n = 4.
Suppose TK = T. Then Li :S: Ki E C(K, T) with Ki/02(Ki) ~ L3(2). But
now KiT E Hz, a case already eliminated. Thus TK < .T, so we have seen that
L]_TK: is the middle-node parabolic.
Let W be a maximal H-submodule of UH, so that UH is irreducible. As
u; E Q(H, UH) and TK: < T, B.4.2 and B.4.5 say that either m(U H) = 6, or U is
the sum of a natural K-module and its dual. The latter is impossible, as L]_TK: is