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 language||English (US)|
|Title of host publication||Progress in Mathematics|
|Number of pages||27|
|State||Published - 2012|
|Name||Progress in Mathematics|
Bibliographical noteFunding Information:
We thank Daniel Bump for helpful conversations and for assistance in preparing Fig. 2.1, which was made using SAGE . 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).
© Springer Science+Business Media, LLC 2012.
- Crystal graph
- Gelfand-Tsetlin pattern
- Littelmann polytope
- Weyl group multiple Dirichlet series
- Whittaker function