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)


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.


Weyl Group Eisenstein Series Dirichlet Series Meromorphic Continuation Stability Assumption 
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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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