CHAPTER 1

NOTATION AND PRELIMINARIES

1.1. Sp2n(F)

The purpose of this chapter is to introduce notation and review some

preliminary facts that will be of use in the rest of this thesis.

Let F be an nonarchimedean local field of characteristic zero. Let O denote

the ring of integers, V the prime ideal in O and w a uniformizer. Then, O jV is a

finite field. Let q denote the number of elements in OjV. We normalize the Haar

measure on F so that vol((D)=l.

Suppose that \ 1S a quasicharacter of JPX (i.e., a multiplicative homomor-

phism from Fx to C x - not necessarily unitary). If x £ Fx, x may be decom-

posed as x = zekxo, with Xo G O*. We can then decompose \ a s X — I \sXu by

X{x) = \zDk\sXu(xo), where \u is a character of (9X, 0 Res 27ri/\nq. It may

be convenient, at times, to view Xu as a character on F* (by Xu(^k^o) = X^(^o))-

We shall use 1 for the trivial character and sgn to denote a nontrivial character

satisying sgn2=l.

As most of this thesis concerns induced representations for Sp2n(F)' we

next discuss Sp2n(F) and induced representations. In this section, we review some

of the structure theory for Sp2n(F).

Recall that we may take

SP2n(F) = {xe GL2n(F)\TXJX = J],

1 Received by the editor Jan. 18, 1991

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