Section 3. Irreducible Modules for Parabolic Subgroups
In the following section s the effec t of various induction functors on
irreducible Pj-modules will be studied. The main object of thi s sectio n i s to
exhibit the parametrization of the irreducible Pj-modules via thei r high (or
low) weights.
The firs t few propositions are well known fact s about Pj-modules, leading
to the determination of the subse t of A which can appear as high weights in a
Pj-module. By the remarks at the end of S2, we need only show the existenc e of
an irreducible Lj-module with the desired high weight. For each weight X in
thi s subset, the irreducible module of high weight X can be constructed from
an irreducible module for the 'semisimple
part1
Lj of Lj.
Let V be a rational Pj-module, and le t A(V) denote the T-weights of V. Then
we have:
Proposition 3.1.
a) Wj stabilize s A(V). More precisely, if w c Wj and X A(V), choose
n w e N G ^ n P = Np(T) i n t n e normalizer of T to represen t w. Then n^*V^ =
Vw(X-
b) Let v. e V^ be a weight vector, and le t a c $ j .
For s a scalar in k we have
0
a
( s ) - v
x
= v
x
+

Z
z +
s
n
v
x + n a
n0
for some weight vector v
x + n a
in v
x + n a
(where 0
a
:k + H U
a
i s the usual root
group isomorphism with the additive group of k; se e [20], for example).
11
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