Symmetric genuine Spherical Whittaker functions on the metaplectic double cover of GSp(2n,F)

TL;DR

This paper provides explicit formulas for symmetric spherical Whittaker functions on the metaplectic double cover of GSp(2n,F), revealing new properties and constructing families of reducible unramified principal series representations.

Contribution

It introduces explicit formulas for symmetric Whittaker functions on metaplectic covers and constructs reducible unramified principal series representations with multiple generic constituents.

Findings

Explicit formulas for spherical Whittaker functions

Symmetry under Weyl group action for these functions

Construction of reducible unramified principal series representations

Abstract

Let F be a p-adic field of odd residual characteristic. Let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F. Let T be a genuine, possibly reducible, unramified principal series representation of G(n). In these notes we give an explicit formulas for a spanning set for the space of Spherical Whittaker functions attached to T. For odd n, and generically for even n, this spanning set is a basis. The signicant property of this set is that each of its elements is unchanged under the action of the Weyl group of G`(n). If n is odd then each element in the set has an equivariant property that generalizes the uniqueness result of Gelbart, Howe and Piatetski-Shapiro proven for G(1). 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.