Fundamental local equivalences in quantum geometric Langlands

Campbell, Justin; Dhillon, Gurbir; Raskin, Sam

doi:10.1112/s0010437x2100765x

Cited by 5 publications

(5 citation statements)

References 30 publications

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“…Let us mention that an important source of both motivation and inspiration for us comes from the following picture. Replacing W ψ ( G, K) with a certain category of twisted sheaves on an affine Grassmannian attached to G, an equivalence to the category of representations of the same quantum group at a root of unity was conjectured by D. Gaitsgory and J. Lurie in [35], described in a form closest in spirit to this work, by Lysenko [72], and studied further in [22,36,37]. In fact it was the Casselman-Shalika formula conjectured (1) Here 'exceptional' in the metaplectic world means that it behaves 'normally,' or as in the linear group case by S. Lysenko [72,Conjecture 11.2.4] linking these twisted sheaves to quantum groups which lies at the origin of this work, and one of our main results is a verification of a version of this conjecture at the level of graded Grothendieck groups.…”

Section: Introductionmentioning

confidence: 78%

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity

Buciumas¹,

Patnaik²

2022

Preprint

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We describe the structure of the Whittaker or Gelfand-Graev module on a n-fold metaplectic cover of a p-adic group G at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of G. To do so, we introduce an algebro-combinatorial model for these modules and develop for them a Kazhdan-Lusztig theory involving new generic parameters. These parameters can either be specialized to Gauss sums to recover the p-adic theory or to the natural grading parameter in the representation theory of quantum groups. As an application of our results, we deduce geometric Casselman-Shalika type results for metaplectic covers, conjectured in a slightly different form by S. Lysenko, as well as prove a variant of G. Savin's local Shimura type correspondences at the Whittaker level.

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Section: Introductionmentioning

confidence: 78%

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity

Buciumas¹,

Patnaik²

2022

Preprint

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“…We can define a highest weight category structure on with standards and costandards . It is the highest weight category structure in [CDR21]. However, it is different from that used in this paper.…”

Section: Standard Object and Duality Of Whittaker Categorymentioning

confidence: 99%

“…KL94], Rep q ( Ǧ), the category of representations of the quantum group, is equivalent to the category of Ǧ(O)-integrable Kac-Moody representations. Furthermore, by [CDR21], the latter is equivalent to the twisted Whittaker category on the affine Grassmannian Gr G := G(K)/G(O). Hence, there is an equivalence of categories Whit(D-mod κ (Gr G )) := D-mod κ (N (K), χ\Gr G ) Rep q ( Ǧ).…”

Section: Conjecture 2 (Iwahori Fundamental Local Equivalence) There I...mentioning

confidence: 99%

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Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Yang

2024

Compositio Math.

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Let $G$ be a reductive group, and let $\check {G}$ be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags ${\operatorname {Fl}}_G$ is equivalent to the category of $\check {G}$ -equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on ${\operatorname {Fl}}_G$ and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category $\mathsf {O}$ is equivalent to the twisted Whittaker category on ${\operatorname {Fl}}_G$ in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on ${\operatorname {Fl}}_G$ and a factorization module category, which holds in the de Rham setting, the Betti setting, and the $\ell$ -adic setting.

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“…The following linkage principle was proposed by Yakimov and will be proven in this section. To do so, one works also with (B, λ)-equivariant objects and the corresponding twisted affine Hecke categories, where λ varies over character sheaves pulled back from the abstract Cartan; see for example [LY20] or [CDR20] for further discussion of such Hecke categories.…”

Section: The Linkage Principle For Positive Energy Representationsmentioning

confidence: 99%

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

Campbell,

Dhillon

2021

Preprint

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We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an integrality hypothesis, we establish monoidal equivalences between blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke category. To do so, we introduce a new singular localization onto Whittaker flag manifolds. The argument for the latter specializes to a somewhat alternative treatment of regular localization, and identifies the category of Lie algebra representations with a generalized infinitesimal character as the parabolic induction, in the sense of categorical representation theory, of a Steinberg module. Contents 1. Introduction 1 2. Preliminaries and notation 8 3. Hecke algebras and invariant vectors 15 4. A category of Lie algebra representations 20 5. Localization I: The sign representation and the Steinberg module 22 6. parabolic induction for groups and Hecke categories 26 7. Localization II: the parabolic induction of the Steinberg module and Lie algebra representations 28 8. The linkage principle for positive energy representations 32 9. Some applications 38 10. Affine Harish-Chandra bimodules 40 References 43

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Fundamental local equivalences in quantum geometric Langlands

Cited by 5 publications

References 30 publications

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity

Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

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