Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

Abstract

We show that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of ${\rm GL}_{r+1}$ can be described as multiple Dirichlet series in $r$ complex variables, whose coefficients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on “string data” previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reflections. The coefficients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

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Authors

Ben Brubaker

Massachusetts Institute of Technology
Cambridge, MA

Daniel Bump

Stanford University
Stanford, CA

Solomon Friedberg

Boston College
Chestnut Hill, MA