TY - CHAP

T1 - Twisted weyl group multiple dirichlet series

T2 - The stable case

AU - Brubaker, Ben

AU - Bump, Daniel

AU - Friedberg, Solomon

PY - 2008/1/1

Y1 - 2008/1/1

N2 - 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.

AB - 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.

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U2 - 10.1007/978-0-8176-4639-4_1

DO - 10.1007/978-0-8176-4639-4_1

M3 - Chapter

AN - SCOPUS:84924815636

T3 - Progress in Mathematics

BT - Progress in Mathematics

PB - Springer Basel

ER -