We begin the Introduction with a short description of the problem to be
considered in this paper. Let F be a p-adic field with char F=0 and odd residual
characteristic; O its ring of integers. Set G = Sp2n(F). Let P = MU be a maximal
parabolic subgroup of G, so M = Glk(F) x Sp2(n-k)(F)' r some fc, 1 h n.
A (quasi)character on M is of the form x ° ^etf, X £ -Fx a n d m a Y be extended
trivially to P. We also denote the resulting characters as x- It will be helpful
to decompose x a s X I ' |5 Xu, with Xu coming from a character of Ox. The
question we investigate is whether n =IndpX 1S irreducible or not, where induction
is normalized so that unitary representations induce to unitary representations. In
the following paragraphs, we describe some techniques for attacking this problem.
In his unpublished notes on admissible representations [Cas 2], Casselman
used Hecke algebras to determine reducibility for ShiF) = Sp2(F) with x u n ~
ramified. In his thesis, Gustafson [Gus] extended these techniques to the maximal
parabolic in Sp2n(F) with Levi factor M = Gln(F), x unramified. In the second
chapter of this thesis (the first consisting of notation and preliminaries), we apply
these Hecke algebra methods to determine reducibility for the maximal parabolic in
Sp2n(F) with Levi factor M = Fx x Sp2(n-i)(F) and arbitrary x- In the third and
fourth chapters, we use a technique of Tadic [Tad 2] involving Jacquet modules to
obtain some general results, as well as some specific ones for small n.
We now describe how Hecke algebras are used to show irreducibility. Let
1x7 be a uniformizer of F. Then, Oj{wO) is a finite field which we denote by F
To the parabolic subgroup P , we may associate a parahoric subgroup B by taking
P(Fg) and lifting it back to Sp2n(F). To the character x on P, we can associate an
open compact subgroup Bx contained in B which has the following property:
*) every subquotient of 7r is generated by its Bx fixed vectors.
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