Ben Brubaker

Metaplectic Whittaker Functions and Crystals of Type B

The spherical metaplectic Whittaker function on the double cover of Sp (2r, F), where F is a nonarchimedean local field, is considered from several different points of view. Previously, an expression, similar to the Casselman–Shalika formula, had been given by Bump, Friedberg, and Hoffstein as a sum is over the Weyl group. It is shown that this coincides with the expression for the p-parts of Weyl group multiple Dirichlet series of type B r as defined by the averaging method of Chinta and Gunnells. Two conjectural expressions as sums over crystals of type B are given and another as the partition function of a free-fermionic six-vertex model system.

On Kubota's Dirichlet series

Abstract Kubota [T. Kubota, Some results concerning reciprocity law and real analytic automorphic functions, in: 1969 Number Theory Institute (Proc. Sympos. Pure Math. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I. (1971), 382–395.] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hostein, Weyl group multiple Dirichlet series I, preprint, http://sporadic.stanford.edu/bump/wmd.pdf.]. Closely related results are in Eckhardt and Patterson [C. Eckhardt and S. J. Patterson, On the Fourier coefficients of biquadratic theta series, Proc. London Math. Soc. (3) 64(2) (1992), 225–264.].

Twisted Weyl Group Multiple Dirichlet Series: The Stable Case

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg, and Hoffstein [2]. Brubaker, Bump, and Friedberg [4] provided for when n is sufficiently large; the coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide with Whittaker coefficients of metaplectic Eisenstein series, but they are studied in these papers by a method that is independent of this fact. The assumption that n is large is called stability and allows a simple description of the Dirichlet series. “Twisted” Dirichet series were introduced in Brubaker, Bump, Friedberg, and Hoffstein [5] without the stability assumption, but only for root systems of type A{inr}. Their description is given differently, in terms of Gauss sums associated to Gelfand-Tsetlin patterns. In this paper, we reimpose the stability assumption and study the twisted multiple Dirichlet series for general Φ by introducing a description of the coefficients in terms of the root system similar to that given in the untwisted case in [4]. We prove the analytic continuation and functional equation of these series, and when Φ = A{inr} we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for the stable case.

Nonvanishing twists of GL(2) automorphic L-functions

Let π be a cuspidal automorphic representation of GL(2,AK). Given any prime integer n, suppose there exists a single nonvanishing nth-order twist of the L-series associated to π at the center of the critical strip. We use the method of multiple Dirichlet series to establish that there exist infinitely many such nonvanishing nth-order twists of the L-series of the representation at the center.

Hecke modules from metaplectic ice

We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of p-adic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.

On Hamiltonians for six-vertex models

In this paper, we explain a connection between a family of free-fermionic six-vertex models and a discrete time evolution operator on one-dimensional Fermionic Fock space. The family of ice models generalize those with domain wall boundary, and we focus on two sets of Boltzmann weights whose partition functions were previously shown to generalize a generating function identity of Tokuyama. We produce associated Hamiltonians that recover these Boltzmann weights, and furthermore calculate the partition functions using commutation relations and elementary combinatorics. We give an expression for these partition functions as determinants, akin to the Jacobi-Trudi identity for Schur polynomials.

Chapter One. Type A Weyl Group Multiple Dirichlet Series

We begin by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, we choose the following parameters.