A crystal definition for symplectic multiple Dirichlet series
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Brubaker_A crystal definition.pdf
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Author(s)
Beineke, Jennifer
Brubaker, Benjamin Brock
Frechette, Sharon
Date Issued
July 2012
Journal
Progress in Mathematics, vol. 300, 2012
Publisher
Birkhauser Boston
Citation
Beineke, Jennifer, Ben Brubaker and Sharon Frechette. "A crystal definition for symplectic multiple Dirichlet series." Chapter 2 in Multiple Dirichlet Series, L-functions and Automorphic Forms, Eds, Daniel Bump, Solomon Friedberg, and Dorian Goldfeld, Springer Science+Business Media, LLC 2012 (Series: Progress in Mathematics, vol. 300. July 31, 2012).
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Author's final manuscript
Abstract
This paper presents a definition for a family of Weyl group multiple Dirichlet series (henceforth \MDS") of Cartan type C using a combinatorial model for crystal bases due to Berenstein-Zelevinsky [2] and Littelmann [12]. Recall that a Weyl group MDS is a Dirichlet series in several complex variables which (at least conjecturally) possesses analytic continuation to a meromorphic function and satisfies functional equations whose action on the complex space is isomorphic to the given Weyl group. In [1], we presented a definition for such a series in terms of a basis for highest weight representations of Sp(2r;C) { Type C Gelfand-Tsetlin patterns { and proved that the series satisfied the conjectured analytic properties in a number of special cases. Here we recast that definition in the language of crystal bases and find that the resulting MDS, whose form appears as an unmotivated miracle in the language of Gelfand-Tsetlin patterns, is more naturally defined in this new language.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
http://www.springer.com/birkhauser/mathematics/book/978-0-8176-8333-7
