Metaplectic Iwahori Whittaker functions and supersymmetric lattice models

Abstract

In this paper we compute new values of Iwahori Whittaker functions on $n$-fold metaplectic covers $\tilde{G}$ of $\mathbf{G}(F)$ with $\mathbf{G}$ a split reductive group over a non-archimedean local field $F$. For every Iwahori Whittaker function $\varphi$, and for every $g\in \tilde{G}$, we evaluate $\varphi (g)$ by recurrence relations over the Weyl group using novel "vector Demazure-Whittaker operators." The general formula and strategy of proof are inspired by ideas appearing in the theory of integrable systems. Specializing to the case of $\mathbf{G}={\mathbf{GL}}_r$, we construct a solvable lattice model of a new type associated with the quantum affine super group $U_q(\widehat{\mathfrak{gl}}(r|n))$ and prove that its partition function equals $\varphi (g)$. To prove this equality we match the recurrence relations on the lattice model side (obtained from the Yang-Baxter equation) to the recurrence relations for $\varphi (g)$ derived by using the representation theory of $\tilde{G}$. Remarkably, there is a bijection between the boundary data specifying the partition function and the data determining all values of the Whittaker functions.

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