Yang-Baxter Equations for General Metaplectic Ice

TL;DR

This paper extends the connection between quantum groups and Whittaker functions on metaplectic covers of $GL_r(F)$, providing new solutions to Yang-Baxter equations for all such covers and linking them to quantum superalgebras.

Contribution

It generalizes previous results to all metaplectic covers of $GL_r(F)$, introducing new Yang-Baxter solutions tied to quantum groups and superalgebras.

Findings

New solutions to Yang-Baxter equations for all metaplectic covers.

Connections established between these solutions and quantum superalgebras.

Extension of prior work from specific to all metaplectic covers.

Abstract

In this paper, we extend results connecting quantum groups to spherical Whittaker functions on metaplectic covers of $GL_r(F)$, for $F$ a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic $n$-fold cover of $GL_r(F)$ a set of Yang-Baxter equations model the action of standard intertwiners on principal series Whittaker functions. These equations arise from a Drinfeld twist of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\frak{gl}}(n)),$ where $v = q^{-1}$ for $q$ the cardinality of the residue field. We extend their results to all metaplectic covers of $GL_r(F)$, providing new solutions to Yang-Baxter equations matching the scattering matrix for the associated Whittaker functions. Each cover has an associated integer invariant $n_Q$ and the resulting solutions are connected to the quantum group $U_{\sqrt{v}}(\widehat{\frak{gl}}(n_Q))$ and quantum superalgebra $U_{\sqrt{v}}(\widehat{\frak{gl}}(1|n_Q))$.