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Let F be a p-adic field of odd residual characteristic. Let G(n) and G`(n) bethe metaplectic double covers of the general symplectic group and thesymplectic group attached to the 2n dimensional symplectic space over F. Let Tbe a genuine, possibly reducible, unramified principal series representation ofG(n). In these notes we give an explicit formulas for a spanning set for thespace of Spherical Whittaker functions attached to T. For odd n, andgenerically for even n, this spanning set is a basis. The signicant property ofthis set is that each of its elements is unchanged under the action of the Weylgroup of G`(n). If n is odd then each element in the set has an equivariantproperty that generalizes the uniqueness result of Gelbart, Howe andPiatetski-Shapiro proven for G(1). Using this symmetric set, we construct afamily of reducible genuine unramified principal series representations whichhave more then one generic constituent. This family contains all the reduciblegenuine unramified principal series representations induced from a unitary dataand exists only for n even.

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