Communications in Number Theory and Physics

Volume 13 (2019)

Number 1

A Yang–Baxter equation for metaplectic ice

Pages: 101 – 148



Ben Brubaker (School of Mathematics, University of Minnesota, Minneapolis, Mn., U.S.A.)

Valentin Buciumas (School of Mathematics and Physics, University of Queensland, St. Lucia, QLD, Australia)

Daniel Bump (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Nathan Gray (Department of Mathematics, Mount Holyoke College, South Hadley, Massachusetts, U.S.A.)


We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $\mathrm{GL}(r, F)$, where $F$ is a non-archimedean 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 $U_{\sqrt{v}} (\widehat{\mathfrak{gl}} (1\vert n))$, 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 $U_{\sqrt{v}} (\widehat{\mathfrak{gl}} (1\vert n))$, mentioned above.

In the appendix (joint with Nathan Gray) we interpret values of spherical Whittaker functions on metaplectic covers of the general linear group over a nonarchimedean local field as partition functions of two different solvable lattice models. We prove the equality of these two partition functions by showing the commutativity of transfer matrices associated to different models via the Yang–Baxter equation.

This work was supported by NSF grants DMS-1406238 (Brubaker), DMS-1001079, DMS-1601026 (Bump and Buciumas), the Max Planck Institute for Mathematics in Bonn (Buciumas) and ERC grant AdG 669655 (Buciumas).

Received 15 November 2017

Accepted 6 September 2018

Published 29 April 2019