696 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW
"{.1. The cases which must be treated in this part
Recall we are assuming Hypothesis 7.0.2 and the notation established in thediscussion following that Hypothesis in the introduction to this chapter.
Section 3.2 determines the list of possibilities for Lo and V. We first extract the
sublist consisting of those cases where Vis not an FF-module for NaL(V)(AutL 0 (V)).
We begin that deduction, later summarizing the final results in the Table of Propo-
sition 7.1.l.
Recall in the Fundamental Setup that V = (V 0 T) for some member Vo of
Irr+(Lo,R2(Lo),T), while VM := (VM), Mv := NM(V), and Mv := Mv/CMv(V)
= Auta(V). We wish to determine the cases where V is not an FF-module for
NaL(V)(AutL 0 (V)).We first consider the case where T </:. Na(L). Here 3.2.6 applies, and we see
that in cases (1) and (2) of 3.2.6, Vis not an FF-module and VM = V =Vo; these
examples appear as the last two cases (below the second horizontal line) in the
Table of Proposition 7.1.1. By part (3) of Hypothesis 7.0.2, case (3) of 3.2.6 does
not hold. These are the modules where V =j=. V 0 ; they are treated later in chapter
10 of part 4 in a uniform manner, although some of these examples are FF-modules
and some are not.
Therefore we may assume that TS Nc(L), so Lo= Land (L, T) =LT. Wefirst consider the case where T </:. Na(Vo), so that case (3) of 3.2.5 holds. These
modules satisfy VM = V = V 0 EBV; fort E T-Nr(Vo); the examples with L ~ L4(2)
or L 5 (2) are FF-modules, but the others are not, and so the latter appear as the
second group in the Table (between the horizontal lines).
Thus we are reduced to the case T S Na (V 0 ), so that V = Vo. Furthermore
Cv(L) = 1 as remarked in the introduction to this chapter, so Vis an irreducibleL-module. These cases are listed in 3.2.9, and form the first group in the Table--
except for the first case 3.2.9.1, which is excluded by part (4) of Hypothesis 7.0.2.
This case was handled in part 2 in the "Generic Case", since the unitary groupsU 4 (2n) arise in that case.
This completes the deduction of Proposition 7.1.1.
. We also indicate, in the last two columns of the Table of that result, first
the "shadows" (that .is, groups having such a local configuration but which are not
quasithin or simple), and then the single simple quasithin example given by J4.
Three of the cases seem to require treatment different from the fairly uniformapproach used to treat the remaining cases. In the final case where V is the orthog-
onal module for Lo= nt(2n), we have m = 2 when n = 2-and worse, a= m = n
for any n, and as Lis not normal in M, we can't appeal to Remark 4.4.2. Because
of these difficulties, this case will be treated by more direct methods in the third
and final chapter of this part. The penultimate case poses similar difficulties, and
is treated in the last section of the second chapter 8 of this part. Finally the case
where Lo'i' ~ Aut(L3(2)) and Vis the sum 3 EB 3 of the natural and dual module
requires special treatment, particularly as m = 2 makes it difficult to establishlemma 7.3.2. This case is dealt with at the end of chapter 7.
We have established the list of cases to be treated under Hypothesis 7.0.2:
PROPOSITION 7.1.l. The cases where Vis not an FF-rq,odule, and which appear