Constructing Weyl group multiple Dirichlet series

Chinta, Gautam; Gunnells, Paul E.

doi:10.1090/s0894-0347-09-00641-9

Cited by 62 publications

(148 citation statements)

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“…We then give examples of such schema coming from the various aforementioned sources: from representations of p-adic groups or their covers, from models for various functionals on these representations, and from R-matrices of quantum groups. We explain how these examples are related and use the formalism to extend the work of [7] to the metaplectic setting, where metaplectic versions of Demazure operators as in Chinta, Gunnells and Puskás [20], using the action of Chinta and Gunnells [19], appear naturally and inherit an R-matrix interpretation.We now describe our results in more detail, first introducing the affine Hecke algebra. Let W be a finite Weyl group, acting on the root system Φ ∨ inside Λ.…”

mentioning

confidence: 99%

Hecke modules from metaplectic ice

Brubaker

Buciumas

Bump

et al. 2017

Sel. Math. New Ser.

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Abstract. We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of padic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.An important tool in studying representations of p-adic groups is the affine Hecke algebra. Ion [28], Brubaker, Bump and Licata [8], Brubaker, Bump and Friedberg [7,9] and other papers considered representations of Hecke algebras related to models of representations of p-adic groups. These works mainly focused on unique functionals such as the Whittaker, spherical and Bessel functionals, but, as we shall show here, the ideas are also applicable to certain non-unique functionals on central extensions of p-adic groups. The goal is to show that these functionals act as Hecke algebra module maps to various target spaces. This results in recursion formulas for Iwahori fixed vectors in the model using Demazure-like operators based on the Hecke action. It allows for effective computation, provides links to the geometry of associated Bott-Samelson varieties, and proves that these functions match important classes of symmetric functions, such as various specializations of non-symmetric Macdonald polynomials.In a separate direction, Brubaker, Bump and Friedberg [6] showed that the values of the spherical Whittaker function for unramified principal series of GL r (F ), where F is a p-adic field, may be realized as partition functions of a solvable lattice model. A similar model for the spherical Whittaker functions for covers of GL r (F ) -a situation in which the Whittaker functional is no longer unique -was given in Brubaker, Bump, Chinta, Friedberg and Gunnells [5]. However, except when the cover is trivial, this model was not solvable in the sense of Baxter [3]. The Boltzmann weights were altered to obtain a solvable model in Brubaker, Buciumas and Bump [4]. This is the "metaplectic ice" of the title. Moreover, [4] provided a new algebraic framework, relating the models to the R-matrices of a Drinfeld twist of the quantum affine Lie superalgebra U q ( gl(n|1)). As a biproduct of this discovery, a relationship between the scattering matrices of the intertwining integrals on Whittaker coinvariants and R-matrices of a Drinfeld twist of U q ( gl(n)) came to light.In the present paper, we will unify these directions by providing simultaneous foundations for both. The search for new foundations is motivated by the following consideration. In [7], we took the point of view of the universal principal series. This means that the induced character becomes C[Λ]...

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confidence: 99%

Hecke modules from metaplectic ice

Brubaker

Buciumas

Bump

et al. 2017

Sel. Math. New Ser.

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“…When n > 1, the Weyl group action on functions is non-obvious; the simple reflections involve Gauss sums and congruence conditions, and verifying the braid relations is not a simple matter. See Chinta and Gunnells [27] for this action, and Patterson [52] for a meditation on the relationship between the method and the intertwining operators for principal series representations.…”

Section: Chinta-gunnellsmentioning

confidence: 99%

“…Let V = V z be as in (27). Let Ω be one of the following two linear functionals on V : it is either the Whittaker functional…”

Section: Demazure Operatorsmentioning

confidence: 99%

Introduction: Multiple Dirichlet Series

Bump

2012

Multiple Dirichlet Series, L-Functions and Automorphic Forms

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“…The first of these is expressed in terms of a Weyl group action described in [10], the second in terms of a function on Gelfand-Tsetlin patterns initially introduced in [7]. In fact, this latter representation belongs to a family of explicit formulas, one for each reduced expression of the long element of the Weyl group as a product of simple reflections.…”

Section: Introductionmentioning

confidence: 99%

“…We give two examples of this: first, the equivalence of two different Gelfand-Tsetlin definitions, and second, the effect of the Weyl group action on the Langlands parameters. The second example is closely connected with another construction of the metaplectic Whittaker function by averaging over a Weyl group action [9,10]. …”

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confidence: 99%

Metaplectic Ice

Brubaker¹,

Bump²,

Chinta³

et al. 2012

Multiple Dirichlet Series, L-Functions and Automorphic Forms

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Abstract. We study spherical Whittaker functions on a metaplectic cover of GL(r + 1) over a nonarchimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker function was given in terms of Gelfand-Tsetlin patterns in [5,17], and we translate this description into an expression of the values of the Whittaker function as partition functions of a sixvertex model. Properties of the Whittaker function may then be expressed in terms of the commutativity of row transfer matrices potentially amenable to proof using the Yang-Baxter equation. We give two examples of this: first, the equivalence of two different Gelfand-Tsetlin definitions, and second, the effect of the Weyl group action on the Langlands parameters. The second example is closely connected with another construction of the metaplectic Whittaker function by averaging over a Weyl group action [9,10].

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Constructing Weyl group multiple Dirichlet series

Abstract: Let Φ \Phi be a reduced root system of rank r r . A Weyl group multiple Dirichlet series for Φ \Phi is a Dirichlet series in r r complex variables s 1 , … , s r s_1,\dots ,s_r , initially converging for R e ( s i ) \mathrm {Re}(s_i) sufficiently l… Show more

Cited by 62 publications

References 28 publications

Hecke modules from metaplectic ice

Hecke modules from metaplectic ice

Introduction: Multiple Dirichlet Series

Metaplectic Ice

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