Twisted Weyl Group Multiple Dirichlet Series: The Stable Case
نویسندگان
چکیده
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 [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their 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] in the stable case.
منابع مشابه
Weyl Group Multiple Dirichlet Series IV : The Stable
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 [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide wi...
متن کاملWeyl Group Multiple Dirichlet Series I
Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet s...
متن کاملec 2 00 6 ON THE p - PARTS OF QUADRATIC WEYL GROUP MULTIPLE DIRICHLET SERIES
Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s1, . . . , sr, initially converging for R(si) sufficiently large, which has meromorphic continuation to C and satisfies functional equations under the transformations of C corresponding to the Weyl group of Φ. Two constructions of such series are available, one [1...
متن کاملOn the p-parts of quadratic Weyl group multiple Dirichlet series
Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s1, . . . , sr, initially converging for <(si) sufficiently large, which has meromorphic continuation to C and satisfies functional equations under the transformations of C corresponding to the Weyl group of Φ. Two constructions of such series are available, one [1...
متن کامل2 00 6 ON THE p - PARTS OF QUADRATIC WEYL GROUP MULTIPLE DIRICHLET SERIES
Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s1, . . . , sr, initially converging for R(si) sufficiently large, which has meromorphic continuation to C and satisfies functional equations under the transformations of C corresponding to the Weyl group of Φ. Two constructions of such series are available, one [1...
متن کامل