Metaplectic Whittaker Functions and Crystals of Type B

Let n be an integer and let F be a nonarchimedean local field whose characteristic is not a prime dividing n. Let μk be the group of k-th roots of unity in the algebraic closure of F ; we assume that μ2n ⊂ F . Let G be a split, simply-connected semisimple algebraic group over F . We assume that G is actually defined over the ring o of integers in F in such a way that K = G(o) is a special maximal compact subgroup of G(F ). Matsumoto [20] constructed an n-fold metaplectic cover G̃(F ) of G(F ). For this, we only need μn ⊂ F but the hypothesis μ2n ⊂ F simplifies the metaplectic cocycle and the resulting formulas. We are interested in values of a spherical Whittaker function W on G̃(F ). Let G = Sp2r and let the cover degree n = 2. In this case, we connect a known description of the Whittaker function to the theory of multiple Dirichlet series: