Some geometric facets of the Langlands correspondence for real groups

Virk, Rahbar

doi:10.1112/blms/bdu105

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…By theorem 1.12 in [LV83] their cohomologies are either concentrated in even degree or odd degree. See the comments in the proof to corollary 4.6 in [Vir15] for example for why this applies to C in addition to F p .…”

Section: It Is Well Known That This Mapsmentioning

confidence: 99%

Sato-Tate Equidistribution for Families of Automorphic Representations through the Stable Trace Formula

Dalal

2019

Preprint

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In [ST16], Shin and Templier proved certain equidistribution bounds on local components of certain families of automorphic representations. We extend their weight-aspect results to families of automorphic representations where the Archimedean component is restricted to a single discrete-series representation instead of an entire L-packet. We do this by using a so-called "hyperendoscopy" version of the stable trace formula developed by Ferrari.The main technical difficulties are defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and bounding the values of transfers of unramified functions. We also present an extension of Arthur's simple trace formula for test functions with Euler-Poincaré component at infinity to non-cuspidal groups since it does not seem to appear elsewhere in the literature.

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Section: It Is Well Known That This Mapsmentioning

confidence: 99%

Sato-Tate Equidistribution for Families of Automorphic Representations through the Stable Trace Formula

Dalal

2019

Preprint

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“…Section III.3, these motives generate all equivariant mixed Tate motives, establishing the equivariant Whitney-Tate condition. The changes necessary to go from [SW15] to the more general equivariant setting mostly follow the approach outlined in [Vir14].…”

Section: Introductionmentioning

confidence: 99%

Equivariant motives and geometric representation theory. (with an appendix by F. Hörmann and M. Wendt)

Soergel,

Virk,

Wendt

2018

Preprint

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We consider categories of equivariant mixed Tate motives, where equivariant is understood in the sense of Borel. We give the two usual definitions of equivariant motives, via the simplicial Borel construction and via algebraic approximations of it. The definitions turn out to be equivalent and give rise to a full six-functor formalism. For rational étale motives over a finite field or the homotopical stable algebraic derivator arising from semisimplified Hodge realization, the equivariant mixed Tate motives provide a graded version of the equivariant derived category. We show that, in sufficiently nice and clean cases, these categories admit weight structures; moreover, a tilting result holds which identifies the category of equivariant mixed Tate motives with the bounded homotopy category of the heart of its weight structure. This can be seen as a formality result for equivariant derived categories. We also discuss convolution functors on equivariant mixed Tate motives, and consequences for the categorification of the Hecke algebras and modules.

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“…We hope that our explanations contribute to a better understanding of the original vision laid out in the work [BG86] of Beilinson and Ginzburg. We also expect that the use of mixed motivic categories will turn out to be fruitful in a lot of other instances where geometric representation theory relies on "mixed geometry". For example, in a joint work in progress with Rahbar Virk, we will discuss how Borel-equivariant motives can be used to establish motivic versions of the results from [Vir13] and construct a very natural geometric categorification of the Hecke algebra.…”

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confidence: 99%

Perverse Motives and Graded Derived Category

Soergel¹,

Wendt²

2016

J. Inst. Math. Jussieu

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For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finitedimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category O and Koszul duality.

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Some geometric facets of the Langlands correspondence for real groups

Cited by 4 publications

References 16 publications

Sato-Tate Equidistribution for Families of Automorphic Representations through the Stable Trace Formula

Sato-Tate Equidistribution for Families of Automorphic Representations through the Stable Trace Formula

Equivariant motives and geometric representation theory. (with an appendix by F. Hörmann and M. Wendt)

Perverse Motives and Graded Derived Category

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