Weyl group multiple Dirichlet series, Eisenstein series and crystal bases


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.


Ben Brubaker

Massachusetts Institute of Technology
Cambridge, MA

Daniel Bump

Stanford University
Stanford, CA

Solomon Friedberg

Boston College
Chestnut Hill, MA