A crystal definition for symplectic multiple Dirichlet series

Abstract

This paper presents a definition for a family of Weyl group multiple Dirichlet series (henceforth \MDS") of Cartan type C using a combinatorial model for crystal bases due to Berenstein-Zelevinsky [2] and Littelmann [12]. Recall that a Weyl group MDS is a Dirichlet series in several complex variables which (at least conjecturally) possesses analytic continuation to a meromorphic function and satisfies functional equations whose action on the complex space is isomorphic to the given Weyl group. In [1], we presented a definition for such a series in terms of a basis for highest weight representations of Sp(2r;C) { Type C Gelfand-Tsetlin patterns { and proved that the series satisfied the conjectured analytic properties in a number of special cases. Here we recast that definition in the language of crystal bases and find that the resulting MDS, whose form appears as an unmotivated miracle in the language of Gelfand-Tsetlin patterns, is more naturally defined in this new language.