PARABOLIC SUBGROUPS ANDINDUCTION

15

R = {(t,t ) t € Lj n T} acts trivially on this, then we will have an induced

action of Lj = (Lj X T)/R on S(e1(X)) $ e2(X), making it into an irreducible

Lj-module. Let v be a high weight vector in S(c1(X)) and let w generate the

1-dimensional module c2(X). Then if t e Lj n T we have {tft_1)»(v $ w) =

t»v ® t^- w = e^XMtJv ® e2(X)(t"1)w = c1(X)(t)c2(X)(t"1)(v ® w)

X(t)X(t"1)v $ w = v 0 w (G1(X) and e2(X) agree on Lj n T, since they are both

restrictions of the character X of T). Now S(e1(X)) $ e2(X) is generated by

v $ w as an L, X T-module so Lj n T acts trivially on all of

Sfe^X)) 0 e2(X).

Q

Lemma 3.5. Any two irreducible Pj-modules with the same high weight are

isomorphic.

Proof. Let v =

(vi'v2^ i n v

i ®

V2' w n e r e v

i

i s a

high weight vector in (

v

jh. Let

V' be the cyclic Pj-submodule of V1 e V2 generated by v. If V' n V^ * 0, it

would be Vjj by irreducibility. In this case (v^O) (respectively, (0,v2)) is a

weight vector of V/ linearly independent with v contradicting the fact that

V/ = kv (Proposition 3.2). Hence we have V' n VA = 0 so the projections

K*:V' - V* are isomorphisms. _

Corollary 3.6. For any X € A+ there exists a unique irreducible Pj-module

Sj(X) of high weight X, and this is a complete list of irreducibles for Pj.

Proof. Existence is Lemma 3.4, while uniqueness is Lemma 3.5. The list is

complete by corollary 3.3.

n

Recall that the Levi factor of a CPS subgroup Hj; is also of the form Lj,

where I = K n J . Thus we also can parametrize the irreducible HjJ-modules by

Aj.

We close this section with a few remarks on low weights and dual modules.

Let Wj represent the long word of Wj and write X = e^X) + e2(X). Then Wj

fixes E^ and so X J = e^X) J + e2(X). For a € J, e2(X),a = 0 so X J,a =