Introduction: Multiple Dirichlet Series

Bump, Daniel

doi:10.1007/978-0-8176-8334-4_1

Cited by 13 publications

(16 citation statements)

References 44 publications

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“…We recall now briefly the Chinta-Gunnells Weyl group action, referring the reader to [4,5,9,[14][15][16] and especially the survey [8] for the connection to Weyl group multiple Dirichlet series. Let W be the Weyl group of an irreducible root system , with Coxeter generators {s i } r i=1 corresponding to a choice of simple roots {α i } r i=1 .…”

Section: The Chinta-gunnells Actionmentioning

confidence: 99%

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

Sahi

Stokman

Venkateswaran

2021

Sel. Math. New Ser.

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We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters $$g_i$$ g i , and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters $$g_i$$ g i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters $$g_i$$ g i . It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the $$g_i$$ g i , and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.

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Section: The Chinta-gunnells Actionmentioning

confidence: 99%

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

Sahi

Stokman

Venkateswaran

2021

Sel. Math. New Ser.

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“…In Section 7, we will see that the right-hand side of (4.6) can be rewritten as a sum involving certain modified Schur-type multiple zeta-functions of hook type. Here we mention the observation that the double series Z µ,ν (s s s; (k, )) can be regarded as an analogue of "Weyl group multiple Dirichlet series" in the sense of Bump [4]. The origin of the theory of Weyl group multiple Dirichlet series goes back to the work of D. Goldfeld and J. Hoffstein [5], in which the two-variable Dirichlet series d≥1 L(w, d)d 1/2−2s was studied, where L(w, d) is essentially the Dirichlet L-function attached to the quadratic character χ d .…”

Section: The Second Main Theoremmentioning

confidence: 99%

“…It is to be stressed that in the statement of Theorem 4.1, an analogue of Weyl group multiple Dirichlet series in the sense of D. Bump et al (see [4]) appears. This point will also be discussed in Section 4.…”

Section: Introductionmentioning

confidence: 98%

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

Matsumoto¹,

Nakasuji²

2021

Publ. Math. Debrecen

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“…Over Z θ max we have a canonical section ofL given by the isomorphisms [4], [6] for a survey). 7.2.1.…”

Section: Consider the Inclusion A : Grmentioning

confidence: 99%

“…The purpose of this section is to establish some results about the Fourier transform of IC Z θ c , ζ over Žθ E,c . This is important in view of a relation with the theory of Weyl group multiple Dirichlet series (see [4], [6] for a survey). 7.2.1.…”

Section: Denote By žθmentioning

confidence: 99%

Geometric Eisenstein series: twisted setting

Lysenko¹

2017

J. Eur. Math. Soc.

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Abstract. Let G be a simple simply-connected group over an algebraically closed field k, X be a smooth connected projective curve over k. In this paper we develop the theory of geometric Eisenstein series on the moduli stack Bun G of G-torsors on X in the setting of the quantum geometric Langlands program (forétaleQ ℓ -sheaves) in analogy with [3]. We calculate the intersection cohomology sheaf on the version of Drinfeld compactification in our twisted setting. In the case G = SL 2 we derive some results about the Fourier coefficients of our Eisenstein series. In the case of G = SL 2 and X = P 1 we also construct the corresponding theta-sheaves and prove their Hecke property.

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Introduction: Multiple Dirichlet Series

Cited by 13 publications

References 44 publications

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

Geometric Eisenstein series: twisted setting

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