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Symmetric Genuine Spherical WhittakerFunctions on

Published online by Cambridge University Press:  20 November 2018

Dani Szpruch
Dani Szpruch*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. e-mail: dszpruch@math.purdue.edu
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Abstract

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Let $F$ be a p-adic field of odd residual characteristic. Let $\overline{GS{{p}_{2n}}(F)}$ and $\overline{S{{p}_{2n}}(F)}$ be the metaplectic double covers of the general symplectic group and the symplectic group attached to the $2n$ dimensional symplectic space over $F$, respectively. Let $\sigma$ be a genuine, possibly reducible, unramified principal series representation of $\overline{GS{{p}_{2n}}(F)}$. In these notes we give an explicit formula for a spanning set for the space of Spherical Whittaker functions attached to $\sigma$. For odd $n$, and generically for even $n$, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of $\overline{S{{p}_{2n}}(F)}$. If $n$ is odd, then each element in the set has an equivariant property that generalizes a uniqueness result proved by Gelbart, Howe, and Piatetski-Shapiro.Using this symmetric set, we construct a family of reducible genuine unramified principal series representations that have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for $n$ even.

Keywords

11F85metaplectic groupCasselman Shalika Formula
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 1 , 01 February 2015 , pp. 214 - 240
Copyright
Copyright © Canadian Mathematical Society 2015

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