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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.
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.