Symmetric Genuine Spherical WhittakerFunctions on

Szpruch, Dani

Canadian journal of mathematics, Tome 67 (2015) no. 1, pp. 214-240

Résumé

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.

Détail

DOI : 10.4153/CJM-2013-033-5

Mots-clés : 11F85, metaplectic group, Casselman Shalika Formula