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A Crystal Definition for Symplectic Multiple Dirichlet Series

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Multiple Dirichlet Series, L-functions and Automorphic Forms

Part of the book series: Progress in Mathematics ((PM,volume 300))

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, 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Western New England University, 1215 Wilbraham Road, Springfield, MA, 01119, USA

    Jennifer Beineke

  2. Department of Mathematics, MIT, Cambridge, MA, 02139-4307, USA

    Ben Brubaker

  3. Department of Mathematics and Computer Science, College of the Holy Cross, 1 College Street, Worcester, MA, 01610, USA

    Sharon Frechette

Authors
  1. Jennifer Beineke
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  2. Ben Brubaker
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  3. Sharon Frechette
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Corresponding author

Correspondence to Ben Brubaker .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, 94305-2125, California, USA

    Daniel Bump

  2. , Department of Mathematics, Boston College, Commonwealth Ave. 140, Chestnut Hill, 02467-3806, Massachusetts, USA

    Solomon Friedberg

  3. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

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Beineke, J., Brubaker, B., Frechette, S. (2012). A Crystal Definition for Symplectic Multiple Dirichlet Series. In: Bump, D., Friedberg, S., Goldfeld, D. (eds) Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol 300. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8334-4_2

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