Eisenstein Series and Applications
Volume 258 of the series Progress in Mathematics pp 126
Twisted Weyl Group Multiple Dirichlet Series: The Stable Case
 Ben BrubakerAffiliated withDepartment of Mathematics, MIT
 , Daniel BumpAffiliated withDepartment of Mathematics, Stanford University
 , Solomon FriedbergAffiliated withDepartment of Mathematics, Boston College
Summary
Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the nth 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 nth 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 GelfandTsetlin 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.
 Title
 Twisted Weyl Group Multiple Dirichlet Series: The Stable Case
 Book Title
 Eisenstein Series and Applications
 Pages
 pp 126
 Copyright
 2008
 DOI
 10.1007/9780817646394_1
 Print ISBN
 9780817644963
 Online ISBN
 9780817646394
 Series Title
 Progress in Mathematics
 Series Volume
 258
 Publisher
 Birkhäuser Boston
 Copyright Holder
 Birkhäuser Boston
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 Editors

 Wee Teck Gan ^{(1)}
 Stephen S. Kudla ^{(2)}
 Yuri Tschinkel ^{(3)}
 Editor Affiliations

 1. Department of Mathematics, University of California
 2. Department of Mathematics, University of Toronto
 3. Courant Institute of Mathematical Sciences, New York University
 Authors

 Ben Brubaker ^{(4)}
 Daniel Bump ^{(5)}
 Solomon Friedberg ^{(6)}
 Author Affiliations

 4. Department of Mathematics, MIT, Cambridge, MA, 021394307, USA
 5. Department of Mathematics, Stanford University, Stanford, CA, 943052125, USA
 6. Department of Mathematics, Boston College, Chestnut Hill, MA, 024673806, USA
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