Authors’ abstract: The authors study spherical Whittaker functions on a metaplectic cover of \(\operatorname{GL}(r+1)\) over a non-Archimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker function was given in terms of Gelfand-Tsetlin patterns in [the authors, Ann. Math. (2) 173, No. 2, 1081–1120 (2011; Zbl 1302.11032); P. J. McNamara, Duke Math. J. 156, No. 1, 1–31 (2011; Zbl 1217.22015)], and they 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 be proved using the Yang-Baxter equation. They 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.
For the entire collection see [Zbl 1257.11001].