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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