A crystal definition for symplectic multiple dirichlet series

Jennifer Beineke, Ben Brubaker, Sharon Frechette

Research output: Chapter in Book/Report/Conference proceedingChapter

5 Scopus citations


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;ℂ). 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.

Original languageEnglish (US)
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
Number of pages27
StatePublished - 2012

Publication series

NameProgress in Mathematics
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

Bibliographical note

Funding Information:
We thank Daniel Bump for helpful conversations and for assistance in preparing Fig. 2.1, which was made using SAGE [19]. We also thank Gautam Chinta, Sol Friedberg, and Paul Gunnells for their shared insights. This work was partially supported by NSF grants DMS-0502730 (Beineke), DMS-0702438, and DMS-0844185 (Brubaker).

Publisher Copyright:
© Springer Science+Business Media, LLC 2012.


  • Crystal graph
  • Gelfand-Tsetlin pattern
  • Littelmann polytope
  • Weyl group multiple Dirichlet series
  • Whittaker function


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