Metaplectic Whittaker Functions and Crystal Bases

TL;DR

This paper develops a method to express Whittaker functions on nonlinear group coverings as sums over crystal graphs, linking them to multiple Dirichlet series in type A, and providing new combinatorial insights.

Contribution

It introduces a novel recipe for representing Whittaker functions via crystal bases, connecting representation theory with number theory in a new way.

Findings

Whittaker functions can be expressed as sums over crystal graphs.

In type A, these expressions match known formulas for multiple Dirichlet series coefficients.

The approach bridges algebraic groups, combinatorics, and number theory.

Abstract

We study Whittaker functions on nonlinear coverings of simple algebraic groups over a non-archimedean local field. We produce a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph, and show that in type A, these expressions agree with known formulae for the prime power supported coefficients of Multiple Dirichlet Series.