Metaplectic Ice for Cartan Type C

University of Minnesota Ph.D. dissertation. May 2017. Major: Mathematics. Advisor: Benjamin Brubaker. 1 computer file (PDF); vii, 77 pages.We use techniques from statistical mechanics to give evidence for new formulas for nonarchimedean metaplectic Whittaker functions, arising in the local theory of automorphic forms. We study a particular variation/generalization of the six-vertex model of type C having domain-wall boundary conditions dependent on a given integer partition $\lambda$ of length at most $r$, where $r$ is a fixed positive integer. More precisely, we examine a planar, non-nested, U-turn model whose partition function $Z_{\lambda}$ is related to characters of the symplectic group $\operatorname{Sp} (2r, \mathbb{C})$. We relate certain admissible states of our statistical-mechanical model to metaplectic Eisenstein series (or equivalently metaplectic Whittaker functions). We give a solution to the Yang--Baxter equation for metaplectic Boltzmann weights, which we use to derive two functional equations involving $Z_{\lambda}$, where one equation describes the action on $Z_{\lambda}$ by a short simple root, while the other describes the action by a long simple root. Finally, we give evidence for the conjecture that $Z_{\lambda}$ is a spherical Whittaker function by showing that $Z_{\lambda}$ satisfies the same identities under our solution to the Yang--Baxter equation as the metaplectic Whittaker function under intertwining operators on the unramified principal series of an $n$-fold metaplectic cover of $\operatorname{SO}(2r+1)$, for $n$ odd