## 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 A_{r}. 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_{r} we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for the stable case.

Original language | English (US) |
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Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

DOIs | |

State | Published - 2008 |

Externally published | Yes |

### Publication series

Name | Progress in Mathematics |
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Volume | 258 |

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.