Metaplectic Iwahori Whittaker functions and supersymmetric lattice models
Abstract
In this paper we consider Iwahori Whittaker functions on $n$fold metaplectic covers $\widetilde{G}$ of $\mathbf{G}(F)$ with $\mathbf{G}$ a split reductive group over a nonarchimedean local field $F$. For every element $\phi$ of a basis of Iwahori Whittaker functions, and for every $g\in\widetilde{G}$, we evaluate $\phi(g)$ by recurrence relations over the Weyl group using "vector DemazureWhittaker operators." Specializing to the case of $\mathbf{G} = \mathbf{GL}_r$, we exhibit a solvable lattice model whose partition function equals $\phi(g)$. These models are of a new type associated with the quantum affine super group $U_q(\widehat{\mathfrak{gl}}(rn))$. The recurrence relations on the representation theory side then correspond to solutions to YangBaxter equations for the lattice models. Remarkably, there is a bijection between the boundary data specifying the partition function and the data determining all values of the Whittaker functions.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.15778
 Bibcode:
 2020arXiv201215778B
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Number Theory;
 Mathematics  Quantum Algebra;
 22E50;
 82B23;
 16T25;
 05E05;
 17B37;
 11F70