A Yang-Baxter equation for metaplectic ice

TL;DR

This paper establishes a connection between metaplectic Whittaker functions and solutions to the Yang-Baxter equation, linking quantum groups, statistical mechanics, and number theory in the context of nonarchimedean local fields.

Contribution

It confirms the Yang-Baxter equation governs metaplectic Whittaker functions and identifies it with a quantum affine Lie superalgebra, incorporating Gauss sums via Drinfeld twisting.

Findings

Yang-Baxter equation underpins Whittaker functions

Identifies the Yang-Baxter equation with quantum affine Lie superalgebra

Shows the scattering matrix matches the twisted R-matrix of quantum groups

Abstract

We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter $v$ is specialized to the inverse of the residue field cardinality.) For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted $R$-matrix of the quantum group $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(n))$. This is a piece of the twisted $R$-matrix for $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, mentioned above.