subregular nilpotent representations of lie algebras in prime ...

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are simple GI{modules. It follows that T (indI V ) has a ltration with factors
pr indI(L V ) with L running over the factors in a composition series of E as a
PI{module. Each L V is the direct sum of all pr(I) 0(L V )with 0 running
over representatives for the orbits of WI on X=pX. SoT (indI V ) has a ltration
with factors
pr indI pr(I) 0(L V ) (2)
with L and 0 as above.
Since , is a weight ofE with multiplicity 1, so is each wi , =
wi , wi 2 W ( , ). Therefore our composition series of E as a PI{module
contains exactly one factor, say Ei, such that wi , is a weight ofEi. Now
wi , is an extremal weight of the GI{module Ei since it is an extremal weight
of the G{module E. (The last statement means for all 2 R that either no
wi , + n with n>0isaweight ofE or that no wi , , n with n>0is
aweight ofE or both.) Therefore wi , is conjugate to the highest weight of
Ei under WI and we can take Ei as the simple module E 0 used in the de nition
of T (I) wi . This means that
T (I) wi (V ) ' pr(I)wi (Ei V ):
So indI(T (I) wi V ) is one of the factors as in (2). [Note that indI(T (I) wi V )
=pr indI (T (I) wi V )by Lemma B.6.]
Let us check next that distinct i lead here to distinct factors. Otherwise
we have i 6= j with (Ei; pr(I)wi )=(Ej; pr(I)wj ). This implies wi , 2
WI(wj , ) and wi 2 WI (wj )+pX. The rst property yields wi 2
wj + ZI; since both X=ZR and ZR=ZI have nop{torsion, the second property
implies that wi 2 WI (wj )+pZI = WI;p (wj ). So = wi , is a
weight of the simple GI{module with extremal weight wj , such that + 2
WI (wj ). Now [9], II.7.7 implies that there exists w 2 WI;p with w = and
w (wj )= + = wi . This is a contradiction to the choice of the wi.
We have so far shown that all indI(T (I) wi V ) occur in the ltration of
T (indI V ) with factors as in (2). It remains to be shown that all remaining
factors as in (2) are 0. Using the exactness of pr , of all pr(I) 0, and of the
induction one reduces rst to the case where V is simple and then to the case
where V = Z ;I(w ) for some w 2 WI. That all other factors as in (2) are 0 in
this case will follow ifwe can show that
dim T (indI V )=
rX
i=1
dim indI(T (I) wi V )
for V = Z ;I(w ). In this case indI V ' Z (w ); so T (indI V ) has by B.3
a ltration with factors Z (w 0 )withw 0 running over wStabWp( ) . Similarly,
each T (I) wi V has a ltration with factors Z ;I(w 0 wi )withw 0 wi running
over wStabWI;p( )wi .ThenindI(T (I) wi V ) has a ltration with factors
Z (w 0 wi ) and w 0 as before. Now the claim follows because StabWp( ) is the
disjoint union of the StabWI;p( )wi .