Twisted weyl group multiple dirichlet series: The stable case

Ben Brubaker, Daniel Bump, Solomon Friedberg

Research output: Chapter in Book/Report/Conference proceedingChapter

16 Scopus citations

Abstract

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 Ar. 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 = Ar we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for the stable case.

Original languageEnglish (US)
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
DOIs
StatePublished - 2008
Externally publishedYes

Publication series

NameProgress in Mathematics
Volume258
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

Bibliographical note

Funding Information:
The authors would like to thank the referee for many helpful comments that improved the exposition of this paper. This work was supported by NSF FRG grants DMS-0354662 and DMS-0353964 and by NSA grant H98230-07-1-0015.

Publisher Copyright:
© 2008, Springer Basel. All rights reserved.

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