Symmetric genuine Spherical Whittaker functions on GSp2n(F)¯

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Let F be a p-adic field of odd residual characteristic. Let GSp2n(F)¯ and Sp2n(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 σ be a genuine, possibly reducible, unramified principal series representation of GSp2n(F)¯. In these notes we give an explicit formula for a spanning set for the space of Spherical Whittaker functions attached to σ. 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 Sp2n(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.

Original languageEnglish
Pages (from-to)214-240
Number of pages27
JournalCanadian Journal of Mathematics
Issue number1
StatePublished - 1 Feb 2015
Externally publishedYes


  • Casselman shalika formula
  • Metaplectic group


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