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Twisted Weyl Group Multiple Dirichlet Series: The Stable Case

  • Ben Brubaker
  • Daniel Bump
  • Solomon Friedberg
Part of the Progress in Mathematics book series (PM, volume 258)

Summary

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg, and Hoffstein [2]. Brubaker, Bump, and Friedberg [4] provided for when n is sufficiently large; the coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide with Whittaker coefficients of metaplectic Eisenstein series, but they are studied in these papers by a method that is independent of this fact. The assumption that n is large is called stability and allows a simple description of the Dirichlet series. “Twisted” Dirichet series were introduced in Brubaker, Bump, Friedberg, and Hoffstein [5] without the stability assumption, but only for root systems of type A{inr}. Their description is given differently, in terms of Gauss sums associated to Gelfand-Tsetlin patterns. In this paper, we reimpose the stability assumption and study the twisted multiple Dirichlet series for general Φ by introducing a description of the coefficients in terms of the root system similar to that given in the untwisted case in [4]. We prove the analytic continuation and functional equation of these series, and when Φ = A{inr} we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for the stable case.

Keywords

Weyl Group Eisenstein Series Dirichlet Series Meromorphic Continuation Stability Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  • Ben Brubaker
    • 1
  • Daniel Bump
    • 2
  • Solomon Friedberg
    • 3
  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsBoston CollegeChestnut HillUSA

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