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

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.