## Abstract

Let F be a p-adic field of odd residual characteristic. Let GSp_{2n}(F)¯ and 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, respectively. Let σ be a genuine, possibly reducible, unramified principal series representation of GSp_{2n}(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 Sp_{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.

Original language | English |
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Pages (from-to) | 214-240 |

Number of pages | 27 |

Journal | Canadian Journal of Mathematics |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2015 |

Externally published | Yes |

## Keywords

- Casselman shalika formula
- Metaplectic group

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