Weyl group multiple Dirichlet series of type C

Jennifer Beineke, Benjamin Brubaker and Sharon Frechette

Vol. 254 (2011), No. 1, 11–46

DOI: 10.2140/pjm.2011.254.11

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).

Keywords

Weyl group multiple Dirichlet series, metaplectic group, Eisenstein series, Gelfand–Tsetlin pattern

Mathematical Subject Classification 2000

Primary: 11F70, 11F68

Secondary: 05E10

Milestones

Received: 14 July 2010

Accepted: 28 February 2011

Published: 7 February 2012