Twisted Whittaker models for metaplectic groups

Lysenko, Sergey

doi:10.1007/s00039-017-0403-1

Cited by 11 publications

(23 citation statements)

References 40 publications

(144 reference statements)

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“…One may also ask for connections with other literature such as Weissman [41]. It seems particularly important to understand the relation between our work and the the quantum geometric Langlands program initiated by Lurie and Gaitsgory in [20], and more specifically the relation to the work of Lysenko [31] and Gaitsgory and Lysenko [21]. Remark 1.2.…”

Section: Introductionmentioning

confidence: 92%

A Yang–Baxter equation for metaplectic ice

Brubaker

Buciumas

Bump

et al. 2019

Communications in Number Theory and Physics

109

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We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic n-fold cover of GL(r, F ), where F is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra U √ v ( gl(1|n)), modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter v is specialized to the inverse of the residue field cardinality.)For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted R-matrix of the quantum group U √ v ( gl(n)). This is a piece of the twisted R-matrix for U √ v ( gl(1|n)), mentioned above.2010 Mathematics Subject Classification. Primary 16T25; Secondary 22E50.1 the representation theory of p-adic groups and quantum groups, which should allow one to use techniques from the theory of quantum groups to study metaplectic Whittaker functions. Although we can now prove this directly, we were led to this result by studying lattice models whose partition functions give values of Whittaker functions on a metaplectic cover of GL(r, F ). In [8], it was predicted that a solvable such model should exist; i.e., one for which a solution to the Yang-Baxter equation exists. Such a solvable model has important applications in number theory: it gives easy proofs (in the style of Kuperberg's proof of the alternating sign matrix conjecture) of several facts about Weyl group multiple Dirichlet series [11]. The other main result of this paper is the discovery of a solvable lattice model whose partition function is a metaplectic Whittaker function. Moreover, we relate this solution to an R-matrix for the quantum affine superalgebra gl(1|n). The relation between the two main results follows from the inclusion of (quantum affine) gl(n) into gl(1|n).We now explain these results in more detail. Let G denote an n-fold metaplectic cover of G := GL(r, F ) where the non-archimedean local field F contains the 2n-th roots of unity. Given a partition λ of length r, we will exhibit a system S λ whose partition function equals the value of one particular spherical Whittaker function at s diag(p λ 1 , . . . , p λr ) , where s : GL(r, F ) → G is a standard section.The systems proposed in [8] were generalizations of the six-vertex model. The six-vertex model with field-free boundary conditions was solved by Lieb [30], Sutherland [40] and Baxter [2] and were motivating examples th...

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Section: Introductionmentioning

confidence: 92%

A Yang–Baxter equation for metaplectic ice

Brubaker

Buciumas

Bump

et al. 2019

Communications in Number Theory and Physics

109

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“…The answer in Proposition 7.1 is given in terms of the perverse sheaf IC ΩX θ ,ζ that has been completely described in [1] in terms of cohomologies of a (part of) the quantum sl 2 at a suitable root of unity. This is a manifestation of the phenomenon that cohomology of quantum groups appear in the quantum geometric Langlands program (the quantum groups were brought into the quantum geometric Langlands program in [12,22]).…”

Section: Resultsmentioning

confidence: 98%

“…ii) The perverse sheaf IC ΩX θ ,ζ also appears in [22] under the name L µ ∅ for G = SL 2 , µ = −mα. Note that for n > 1 the so-called subtop cohomology property is satisfied for our metaplectic data for SL 2 by ( [22], Theorem 1.1.6). So, for n > 1 the perverse sheaf IC Here θ = mα, and the sum is over m ≥ 0 such that m + g − 1 ∈ eZ.…”

Section: It Lifts To a Diagram Of Isomorphismsmentioning

confidence: 99%

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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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“…The ℓ-adic setting is truly a geometrization of automorphic forms over function fields, and many constructions were geometrized: Braverman and Gaitsgory geometrized Eisenstein series [BG02], and Lysenko geometrized in particular Rankin-Selberg integrals [Lys02], theta correspondences [Lys04, Lys11, LaLy13], and several constructions for metaplectic groups [Lys15,Lys17].…”

Section: Geometric Langlands Programmentioning

confidence: 99%

Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

Lafforgue

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text).The Langlands correspondence [Lan70] is a conjecture of utmost importance, concerning global fields, i.e. number fields and function fields. Many excellent surveys are available, for example [Gel84,Bum97,BeGe03,Tay04,Fre07,Art14]. The Langlands correspondence belongs to a huge system of conjectures (Langlands functoriality, Grothendieck's vision of motives, special values of L-functions, Ramanujan-Petersson conjecture, generalized Riemann hypothesis). This system has a remarkable deepness and logical coherence and many cases of these conjectures have already been established. Moreover the Langlands correspondence over function fields admits a geometrization, the "geometric Langlands program", which is related to conformal field theory in Theoretical Physics.Let G be a connected reductive group over a global field F . For the sake of simplicity we assume G is split.The Langlands correspondence relates two fundamental objects, of very different nature, whose definition will be recalled later,

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Twisted Whittaker models for metaplectic groups

Cited by 11 publications

References 40 publications

A Yang–Baxter equation for metaplectic ice

A Yang–Baxter equation for metaplectic ice

Geometric Eisenstein series: twisted setting

Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

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