6 Chris Jantzen
The sign oc is used only to indicate that the last factor comes from a symplectic
group. If ki = 0, the representation will be denoted
pi x ...pi-i oc 1.
The trivial representation of Sp2n(F), n 0, will be denoted £rn, so there will be
no confusion.
We close these preliminaries with four theorems on induced representations
and Jacquet modules.
THEOREM 1.2.1 (Probenius Reciprocity) Let G be a connected reductive p-adic
group, P = MU a parabolic subgroup, p an admissible representation of M, rr an
admissible representation ofG. Then,
HomM(MG(),p) = HomG(7r,2GM/)-
Proof cf. [B-Z2].
THEOREM 1.2.2 Let (r, (2, L) be an irreducible subquotient of Indpminifi (ty a char-
acter).Then, there is a w £ W so that r embeds in Indp . wip. Moreover, all
Indp . w%j) for w 6 W have the same components.
Proof See [Cas2] for references for the first part and [B-Z2] for the second.
We next recall the following theorem, which says the constructions of in-
duced representations and Jacquet modules may be done in stages.
THEOREM 1.2.3 Let L M be standard Levis for G. Then
1-
IGL
= iGM °
IML
2. rLG =
TLM
°
TMG-
Proof, cf. [B-Z2].
We close with a theorem of Bernstein-Zelevinsky, Casselman. Let M, N
be standard Levis for a connected reductive p-adic group G. Set WMN = {w 6
W|w(-Pmin H M) C -Pmin5^~1(-Pmin H N) C Pmin}- We remark that these correspond
to the elements of shortest length in the double-cosets WN\W/WM (WM = Weyl
group of M , etc.).
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