Metaplectic Ice for Cartan Type C

We use techniques from statistical mechanics to provide new formulas for Whittaker coefficients of metaplectic Eisenstein series on odd orthogonal groups, matching Friedberg and Zhang. We study a particular variation/generalization of the six-vertex model of Cartan 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 functions $Z_{\lambda}$ are a generalization of a deformation of characters of the symplectic group $\operatorname{Sp}(2r, \mathbb{C})$. Special cases appeared in: Kuperberg; Hamel and King; Brubaker, Bump, Chinta, and Gunnells; Ivanov. Our main result is that these new families of "metaplectic" models are solvable---i.e., they possess Yang--Baxter equations. We use this to derive two types of functional equations involving $Z_{\lambda}$ corresponding to the two root lengths for simple reflections of the symplectic Weyl group. It is widely believed that the local component of metaplectic Eisenstein series is a metaplectic Whittaker function, though this is subtle owing to the lack of uniqueness of Whittaker models and only verified in type A by McNamara. Thus, we also 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.