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).
Lemma 3.5. Any two irreducible Pj-modules with the same high weight are
Proof. Let v =
(vi'v2^ i n v
i ®
V2' w n e r e v
i s a
high weight vector in (
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.
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
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 =
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