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# Symmetric Genuine Spherical Whittaker Functions on $\overline{GSp_{2n}(F)}$

Published:2013-10-12
Printed: Feb 2015
• Dani Szpruch,
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
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## Abstract

Let $F$ be a p-adic field of odd residual characteristic. Let $\overline{GSp_{2n}(F)}$ and $\overline{Sp_{2n}(F)}$ be the metaplectic double covers of the general symplectic group and the symplectic group attached to the $2n$ dimensional symplectic space over $F$. Let $\sigma$ be a genuine, possibly reducible, unramified principal series representation of $\overline{GSp_{2n}(F)}$. In these notes we give an explicit formulas 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 $\overline{Sp_{2n}(F)}$. If $n$ is odd then each element in the set has an equivariant property that generalizes a uniqueness result of Gelbart, Howe and Piatetski-Shapiro. Using this symmetric set, we construct a family of reducible genuine unramified principal series representations which 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: metaplectic group, Casselman Shalika Formula
 MSC Classifications: 11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]