Chapter

Multiple Dirichlet Series, L-functions and Automorphic Forms

Volume 300 of the series Progress in Mathematics pp 37-63

Date:

A Crystal Definition for Symplectic Multiple Dirichlet Series

  • Jennifer BeinekeAffiliated withDepartment of Mathematics, Western New England University
  • , Ben BrubakerAffiliated withDepartment of Mathematics, MIT Email author 
  • , Sharon FrechetteAffiliated withDepartment of Mathematics and Computer Science, College of the Holy Cross

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Abstract

We present a definition for Weyl group multiple Dirichlet series (MDS) of Cartan type C, where the coefficients of the series are given by statistics on crystal graphs for certain highest-weight representations of Sp(2r, ). In earlier work (Beineke et al., Pacific J. Math., 2011), we presented a definition based on Gelfand–Tsetlin patterns, and the equivalence of the two definitions is explained here. Finally, we demonstrate how to prove analytic continuation and functional equations for any multiple Dirichlet series with fixed data by reduction to rank one information. This method is amenable to MDS of all types.

Keywords

Weylgroup multiple Dirichlet series Crystal graph Gelfand-Tsetlin pattern Littelmann polytope Whittaker function