Metaplectic Ice for Cartan Type C
Abstract
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 sixvertex model of Cartan type C having "domainwall 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, nonnested, Uturn 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 solvablei.e., they possess YangBaxter 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 YangBaxter 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1709.04971
 Bibcode:
 2017arXiv170904971G
 Keywords:

 Mathematics  Representation Theory
 EPrint:
 51 pages