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 non-archimedean 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 Demazure-Whittaker 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 \(U_q(\widehat{\mathfrak{gl}}(r|n))\). The recurrence relations on the representation theory side then correspond to solutions to Yang-Baxter equations for the lattice models.