Vol. 254, No. 1, 2011

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Weyl group multiple Dirichlet series of type C

Jennifer Beineke, Benjamin Brubaker and Sharon Frechette

Vol. 254 (2011), No. 1, 11–46

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).

Weyl group multiple Dirichlet series, metaplectic group, Eisenstein series, Gelfand–Tsetlin pattern
Mathematical Subject Classification 2000
Primary: 11F70, 11F68
Secondary: 05E10
Received: 14 July 2010
Accepted: 28 February 2011
Published: 7 February 2012
Jennifer Beineke
Department of Mathematics
Western New England University
Springfield, MA 01119
United States
Benjamin Brubaker
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139-4307
United States
Sharon Frechette
Department of Mathematics and Computer Science
College of the Holy Cross
Worcester, MA 01610
United States