Metaplectic Ice

TL;DR

This paper explores the connection between metaplectic Whittaker functions on certain algebraic groups and statistical mechanical models, revealing new algebraic and combinatorial insights into their properties.

Contribution

It demonstrates that properties of metaplectic Whittaker functions can be expressed via transfer matrices and suggests proof methods using the Yang-Baxter equation.

Findings

Whittaker functions relate to six-vertex model variants

Properties can be expressed through transfer matrix commutativity

Potential proof strategies involve the Yang-Baxter equation

Abstract

Spherical Whittaker functions on the metaplectic n-fold cover of GL(r+1) over a nonarchimedean local field containing n distinct n-th roots of unity may be expressed as the partition functions of statistical mechanical systems that are variants of the six-vertex model. If n=1 then in view of the Casselman-Shalika formula this fact is related to Tokuyama's deformation of the Weyl character formula. It is shown that various properties of these Whittaker functions may be expressed in terms of the commutativity of row transfer matrices for the system. Potentially these properties (which are already proved by other methods, but very nontrivial) are amenable to proof by the Yang-Baxter equation.