A YangBaxter 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 YangBaxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding YangBaxter equation with that of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1n))$, 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 $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1n))$, mentioned above.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 arXiv:
 arXiv:1604.02206
 Bibcode:
 2016arXiv160402206B
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Number Theory;
 Mathematics  Quantum Algebra;
 16T25 (primary) 22E50 (secondary)