Coherent Springer theory and the categorical Deligne-Langlands correspondence

Ben‐Zvi, Dani; Chen, Harrison; Helm, David; Nadler, David A.

doi:10.48550/arxiv.2010.02321

Cited by 4 publications

(8 citation statements)

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“…Recently, the affine Hecke algebra itself has also seen a parallel realization as the Hochschild homology of the Steinberg stack [BCHN22], and by the formalism of iterated traces [BN21, CP19, GKRV20] Hochschild homology has a realization as the self-Exts of a certain coherent Springer sheaf which lives in the categorical Hochschild homology of the equivariant nilpotent cone, i.e. the category CohpLp p N {G _ ˆGm qq of coherent sheaves on the derived loop space of the equivariant nilpotent cone.…”

Section: It Contains the Category Of Coherent Sheaves Cohp Pmentioning

confidence: 99%

“…I would like to thank David Ben-Zvi for extensive discussions on this subject in preparation of the paper [BCHN22], and for many enlightening discussions surrounding shearing and (de)-equivariantization. I would like to thank Gurbir Dhillon for enlightening conversations surrounding equivariant D-modules and dicsussion surrounding the renormalization which appears in the work [CD23].…”

Section: Acknowledgementsmentioning

confidence: 99%

“…For the former map, it is equivalent to the claim that Ω 1 X{Y r1s has vanishing cohomology in positive degrees, which follows by representability. Claim (b) is standard; see for example [BCHN22,Lem. 3.12].…”

Section: 32mentioning

confidence: 99%

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Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

Chen¹

2023

Preprint

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In this paper we establish an equivalence between the category of S 1 -equivariant coherent sheaves on the formal loop space p LX of a smooth stack X and the category of coherent filtered D-modules on X, generalizing results from the case where X is a smooth variety. This equivalence interpolates between categories of coherent sheaves on stacks appearing as categorical traces in algebro-geometric settings, and categories of equivariant constructible sheaves in topological settings. Unlike in the case of varieties, the subcategory of coherent Dmodules on a smooth stack X is larger than the subcategory of compact objects; we identify the corresponding Koszul dual subcategory as the category of S 1 -equivariant ind-continuous coherent sheaves on p LX.

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Section: It Contains the Category Of Coherent Sheaves Cohp Pmentioning

confidence: 99%

Section: Acknowledgementsmentioning

confidence: 99%

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Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

Chen¹

2023

Preprint

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“…This perspective was profoundly developed in the monumental work of Fargues and Scholze [FS21], which in particular establishes the "automorphic-to-Galois" direction, showing that the automorphic category (and thus its subcategory of representations of GpKq) sheafifies over the stack of Langlands parameters. Upcoming work [HZ] of Hemo and Zhu applies the theory of categorical traces (as in [Zh18]) and Bezrukavnikov's tamely ramified local geometric Langlands correspondence [Bez06,Bez16] to establish a coherent local Langlands correspondence for unipotent representations (the principal series part of which is proved in [BCHN22]).…”

Section: Nbmentioning

confidence: 99%

Betti Geometric Langlands

Ben‐Zvi¹,

Nadler²

2018

Algebraic Geometry: Salt Lake City 2015

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Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory (due to Vogan and Soergel) describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands (due to Fargues, Zhu and Hellmann) describe representations with varying central character of a large family of groups associated to isocrystals in terms of coherent sheaves. The latter conjectures also take place on a larger parameter space, in which Frobenius (or complex conjugation) is allowed a unipotent part.In this article we propose a general mechanism that interpolates between these two settings. This mechanism derives from the theory of cyclic homology, as interpreted through circle actions in derived algebraic geometry. We apply this perspective to categorical forms of the local Langlands conjectures for both archimedean and non-archimedean local fields. In the nonarchimedean case, we describe how circle actions relate coherent and constructible realizations of affine Hecke algebras and of all smooth representations of GLn, and propose a mechanism to relate the two settings in general. In the archimedean case, we explain how to use circle actions to derive the constructible local Langlands correspondence (in the form due to Adams-Barbasch-Vogan and Soergel) from a coherent form (a real counterpart to Fargues' conjecture): the tamely ramified geometric Langlands conjecture on the twistor line, which we survey.

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“…Let U n be the unitary group associated to W n and U n−1 the stabilizer of w in (Step 1) The Langlands functoriality in families is considered in several works (e.g. [15]) and seems approachable in the framework considered in [9,16,59].…”

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confidence: 99%

Families of Canonical Local Periods on Spherical Varieties

Cai¹,

Fan²

2021

Preprint

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In this paper, we consider the variation of canonical local periods on spherical varieties proposed by Sakellaridis-Venkatesh in families. We formulate conjectures for the meromorphic property of these canonical local periods. As examples, we discuss these conjectures for the triple product case and the Gan-Gross-Prasad case. 3.4. Families of spherical characters 12 3.5. Homological multiplicities 13 4. The Triple product case 17 4.1. Jacquet-Langlands-Shimizu lifting and PSR zeta integral in families 17 4.2. Triple product local period in families 23 5. The Gan-Gross-Prasad case 24 5.1. Rankin-Selberg and Asai zeta integral in families 24 5.2. GGP spherical character in families 27 Appendix A. Co-Whittaker families 30 Appendix B. Bernstein variety and rigidity of automorphic type 31 References 32

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Coherent Springer theory and the categorical Deligne-Langlands correspondence

Cited by 4 publications

References 50 publications

Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

Betti Geometric Langlands

Families of Canonical Local Periods on Spherical Varieties

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