Weyl group multiple Dirichlet series of typeC

Abstract

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For a 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. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover degree n. For type C and n odd, we construct an infinite family of Dirichlet series and prove they satisfy the above analytic properties in many cases. The coefficients are exponential sums built from Gelfand-Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg required n sufficiently large, so that coefficients were described by Weyl group orbits. We demonstrate that these two radically different descriptions match when both are defined. Moreover, for n = 1, we prove our series are Whittaker coefficients of Eisenstein series on SO(2r + 1).

Introduction: Multiple Dirichlet Series

“…The Dirichlet series is allowed to depend on a parameter Ψ drawn from a finite-dimensional vector space Ω. It is assumed convergent in some region C such as the one in the following figure, which shows the region for (2). We have graphed the projection onto R 2 obtained by taking the real parts of s 1 and s 2 .…”

Eisenstein series on covers of odd orthogonal groups

“…(The notation d λ (α) is the same as that of [8].) Then in view of the remarks above the following Corollary follows from Theorem 5 above together with Theorem 1 of [4].…”

“…Analogously to the definition of a stable Gelfand-Tsetlin pattern in [4], Definition 10, we define a stable BZL-pattern. One may check the following two facts from the definitions.…”

“…Then Proposition 12 of [4] and the following discussion may be recast in the language of stable BZL-patterns as follows.…”

“…For Eisenstein series on odd degree covers of a split orthogonal group of Cartan type B, a conjectural description of these terms using crystal graphs was given by Beineke, Brubaker and Frechette [3,4]. The quantum group that appears should be regarded as attached to the quantized Lie algebra of the L-group.…”