ABSTRACT

A brief description of the problem under consideration follows. Let F

be a p-adic field (we take char F=0, odd residual characteristic), G = Sp2n(F)-

Let P = MU be a maximal parabolic subgroup of (7, and \ a one-dimensional

representation of M. We may extend x trivially to P. The question we investigate

is whether TT =IndpX ® 1 is irreducible or not.

Two different approaches to this problem are used. The first, based on

the work of Casselman and subsequent work by Gustafson, reduces the problem to

the corresponding question about an associated finite-dimensional representation

of a certain Hecke algebra. We use this method to do the case where M — Fx x

Sp2(n-i)(F)- The second approach is based on a technique of Tadic, and involves

an analysis of Jacquet modules. This is used to prove a more general theorem

on induced representations, which may be used to deal with the problem when \

satisfies a regularity condition. We use this method and ad hoc arguments to work

out the low rank cases completely.

key words and phrases: p-adic field, symplectic group, induced representation,

Jacquet module, Hecke algebra.

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