Ben Brubaker — Page 2

arXiv2010

Metaplectic Ice

Spherical Whittaker functions on the metaplectic n-fold cover of GL(r+1) over a nonarchimedean local field containing n distinct n-th roots of unity may be expressed as the partition functions of statistical mechanical systems that are variants of the six-vertex model. If n=1 then in view of the Casselman-Shalika formula this fact is related to Tokuyama's deformation of the Weyl character formula. It is shown that various properties of these Whittaker functions may be expressed in terms of the commutativity of row transfer matrices for the system. Potentially these properties (which are already proved by other methods, but very nontrivial) are amenable to proof by the Yang-Baxter equation.

math.RT↗

arXiv2010

Weyl group multiple Dirichlet series of type C

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. In type C, they conjecturally arise from the Fourier-Whittaker coefficients of minimal parabolic Eisenstein series on an n-fold metaplectic cover of SO(2r+1). For any odd n, we construct an infinite family of Dirichlet series conjecturally satisfying the above analytic properties. The coefficients of these series are exponential sums built from Gelfand-Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg in [6] and [7] required n to be sufficiently large, so that coefficients could be described by Weyl group orbits. We demonstrate that our construction agrees with that of [6] and [7] in the case where both series are defined, and hence inherits the desired analytic properties for n sufficiently large. Moreover our construction is valid even for n=1, where we prove our series is a Whittaker coefficient of an Eisenstein series. This requires the Casselman-Shalika formula for unramified principal series and a remarkable deformation of the Weyl character formula of Hamel and King [20].

math.NT↗

arXiv2010

Schur Polynomials and the Yang-Baxter equation

We show that within the six-vertex model there is a parametrized Yang-Baxter equation with nonabelian parameter group GL(2)xGL(1) at the center of the disordered regime. As an application we rederive deformations of the Weyl character formule of Tokuyama and of Hamel and King.

math.CO↗