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

+

n£

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).

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