We study spherical Whittaker functions on a metaplectic cover of GL(r + 1) over a nonarchimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker function was given in terms of Gelfand-Tsetlin patterns in (Brubaker et al., Ann. of Math. 173(2):1081-1120, 2011; McNamara, Duke Math. J. 156:29-31, 2011), and we translate this description into an expression of the values of the Whittaker function as partition functions of a six-vertex model. Properties of the Whittaker function may then be expressed in terms of the commutativity of row transfer matrices potentially amenable to proof using the Yang-Baxter equation. We give two examples of this: first, the equivalence of two different Gelfand-Tsetlin definitions, and second, the effect of the Weyl group action on the Langlands parameters. The second example is closely connected with another construction of the metaplectic Whittaker function by averaging over a Weyl group action (Chinta and Gunnells, J. Amer. Math. Soc. 23:189-215, 2010; Chinta and Offen, Amer. J. Math., 2011).
Bibliographical notePublisher Copyright:
© 2012 Springer Science+Business Media, LLC. All rights reserved.
- Crystal graph
- Metaplectic group
- Solvable lattice model
- Weyl group multiple dirichlet series
- Whittaker function
- Yang-baxter equation