Coefficients of the n-Fold Theta Function and Weyl Group Multiple Dirichlet Series

  • Benjamin Brubaker
  • Daniel Bump
  • Solomon Friedberg
  • Jeffrey Hoffstein
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 9)

Abstract

We establish a link between certain Whittaker coefficients of the generalized metaplectic theta functions, first studied by Kazhdan and Patterson [Kazhdan and Patterson, Metaplectic forms, Inst. Hautes Études Sci. Publ. Math., (59): 35–142, 1984], and the coefficients of stable Weyl group multiple Dirichlet series defined in [Brubaker, Bump, Friedberg, Weyl group multiple Dirichlet series. II. The stable case. Invent. Math., 165(2):325–355, 2006]. The generalized theta functions are the residues of Eisenstein series on a metaplectic n-fold cover of the general linear group. For n sufficiently large, we consider different Whittaker coefficients for such a theta function which lie in the orbit of Hecke operators at a given prime p. These are shown to be equal (up to an explicit constant) to the p-power supported coefficients of a Weyl group multiple Dirichlet series (MDS). These MDS coefficients are described in terms of the underlying root system; they have also recently been identified as the values of a p-adic Whittaker function attached to an unramified principal series representation on the metaplectic cover of the general linear group.

Notes

Acknowledgements

This work was supported by NSF grants DMS-0844185, DMS-1001079 and DMS-1001326, NSF FRG grants DMS-0652817, DMS-0652609, and DMS-0652312, and by NSA grant H98230-10-1-0183.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Benjamin Brubaker
    • 1
  • Daniel Bump
    • 2
  • Solomon Friedberg
    • 3
  • Jeffrey Hoffstein
    • 4
  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsBoston CollegeChestnut HillUSA
  4. 4.Department of MathematicsBrown UniversityProvidenceUSA

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