A Yang-Baxter equation for metaplectic ice

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.

References in corpus (3)

• Whittaker functions on metaplectic covers of GL(r)
• Metaplectic Ice for Cartan Type C
• Duality for metaplectic ice

Cited by in corpus (10)

• Colored Vertex Models and Iwahori Whittaker Functions
• Metaplectic Ice for Cartan Type C
• Duality for metaplectic ice
• Stochastic symplectic ice
• Izergin-Korepin analysis on the projected wavefunctions of the generalized free-fermion model
• Dual wavefunction of the symplectic ice
• Colored five-vertex models and Demazure atoms
• Distinguished theta representations for certain covering groups
• Scalar products of the elliptic Felderhof model and elliptic Cauchy formula
• Hecke Modules from Metaplectic Ice