Weyl group multiple Dirichlet series constructed from quadratic characters

Chinta, Gautam; Gunnells, Paul E.

doi:10.1007/s00222-006-0014-1

Cited by 34 publications

(110 citation statements)

References 23 publications

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“…For various choices of and a positive integer n, infinite families of Weyl group multiple Dirichlet series defined over any number field F containing the 2n-th roots of unity were introduced in [Chinta and Gunnells 2007;Brubaker et al 2007;2008]. The coefficients of these Dirichlet series are intimately related to the n-th power reciprocity law in F. It is further expected that these families are related to metaplectic Eisenstein series as follows.…”

Section: Introductionmentioning

confidence: 99%

Weyl group multiple Dirichlet series of typeC

2011

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We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For a root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover degree n. For type C and n odd, we construct an infinite family of Dirichlet series and prove they satisfy the above analytic properties in many cases. The coefficients are exponential sums built from Gelfand-Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg required n sufficiently large, so that coefficients were described by Weyl group orbits. We demonstrate that these two radically different descriptions match when both are defined. Moreover, for n = 1, we prove our series are Whittaker coefficients of Eisenstein series on SO(2r + 1).

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Section: Introductionmentioning

confidence: 99%

Weyl group multiple Dirichlet series of typeC

2011

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“…It is very likely that a similar phenomenon accounts for the resemblance for arbitrary n. Regardless, it suggests a promising approach to the problem of defining unstable Weyl group multiple Dirichlet series-namely, to generalize to arbitrary n the invariant function methods used in [6,7,9] to treat the quadratic case. A first step, using a group action motivated by the functional equation (4.4), has been carried out in a joint work with Gunnells, [10].…”

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confidence: 99%

“…In the unstable case, that is, when n is small relative to Φ, a complete description of the coefficients of the Weyl group multiple Dirichlet series does not yet exist except in the case n = 2, which was treated in Chinta and Gunnells [9]. Important partial progress including a beautiful conjecture for multiple Dirichlet series associated to root systems of type A r is given in Brubaker, Bump, Friedberg and Hoffstein [4].…”

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confidence: 99%

Multiple Dirichlet series over rational function fields

Chinta

2008

Acta Arith.

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Multiple Dirichlet series over rational function fields by Gautam Chinta (New York)1. Introduction. The purpose of this paper is to explicitly compute some examples of Weyl group multiple Dirichlet series over the rational function field F q (t). As described in [2], these are Dirichlet series in r complex variables s 1 , . . . , s r whose coefficients can be expressed in terms of nth order Gauss sums. The general theory implies that over a function field, these multiple Dirichlet series will be rational functions of q −s 1 , . . . , q −sr . Except when n = 2, no examples of these rational functions have been written down.Using explicit knowledge of the functional equations, we will express the A 2 series as a rational function of q −s 1 and q −s 2 . This is the main result of this paper and is given in Theorem 4.2. The functional equations of multiple Dirichlet series arise from the functional equations of single variable Gauss sum Dirichlet series of the type initially studied by Kubota [17] using the theory of metaplectic Eisenstein series on the n-fold cover of GL 2 . This theory was further developed by Kazhdan and Patterson [16] who studied Eisenstein series on the n-fold cover of GL r . It is conjectured that the Weyl group multiple Dirichlet series are related to Whittaker coefficients of these metaplectic Eisenstein series. This conjecture and much supporting evidence for it is given in [2,4] In [2] is described a heuristic method to associate to a positive integer n, and a root system Φ of rank r, a multiple Dirichlet series Z in r complex variables with coefficients given by nth order Gauss sums. Moreover, Z is expected to have an analytic continuation to C r and to satisfy a group of functional equations isomorphic to W , the Weyl group of the root system. Brubaker, Bump, and Friedberg [3] have given a precise definition of Z in the stable case; by definition, this means n is sufficiently large for a fixed Φ. In [3] the authors show that for such n, the Weyl group multiple Dirichlet

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“…This is accomplished in the present paper. We define metaplectic Demazure and Demazure-Lusztig operators using the metaplectic Weyl group action found in [CG10,CG07]. We prove that these operators satisfy the same relations as their classical counterparts.…”

Section: Introductionmentioning

confidence: 94%

“…The formulas in these works involve a "metaplectic" action of the Weyl group on rational functions. This action, which has its origins in Kazhdan-Patterson's seminal investigation of automorphic forms on metaplectic covers of GL r [KP84], was used by two of us (GC and PG) to construct Weyl group multiple Dirichlet series [CG07,CG10]. These are infinite series in several complex variables analogous to the classical Dirichlet series in one variable, such as the Riemann ζ function and Dirichlet L-functions.…”

Section: Introductionmentioning

confidence: 99%

Metaplectic Demazure operators and Whittaker functions

Chinta¹,

Gunnells²,

́as³

2017

Indiana Univ. Math. J.

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Abstract. In [CG10] the first two named authors defined an action of a Weyl group on rational functions and used it to construct multiple Dirichlet series. These series are related to Whittaker functions on an n-fold metaplectic cover of a reductive group. In this paper, we define metaplectic analogues of the Demazure and Demazure-Lusztig operators. We show how these operators can be used to recover the formulas from [CG10], and how, together with results of McNamara [McN], they can be used to compute Whittaker functions on metaplectic groups over p-adic fields.

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Weyl group multiple Dirichlet series constructed from quadratic characters

Cited by 34 publications

References 23 publications

Weyl group multiple Dirichlet series of typeC

Weyl group multiple Dirichlet series of typeC

Multiple Dirichlet series over rational function fields

Metaplectic Demazure operators and Whittaker functions

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