Mixed motives and geometric representation theory in equal characteristic

Eberhardt, Jens Niklas; Kelly, Shane

doi:10.1007/s00029-019-0475-x

Cited by 10 publications

(11 citation statements)

References 35 publications

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“…One can construct pointwise pure objects by different methods, for example, (1)using affinely stratified resolutions of singularities of closures of strata in

X

, see [18, Theorem 4.5]; (2)using contracting

\mathbb {G}_m

actions, see [33, Proposition 7.3. ]; (3)by an inductive process in the case of flag varieties, see [33, Lemma 6.6]. All of those methods can be used to show that all objects in

\hbox{D}^{mix}_\mathcal {S}(X)^{w=0}

and

\hbox{DK}_\mathcal {S}(X)^{w=0}

are pointwise pure in the case of flag varieties.…”

Section: Motivic Sheaves On Affinely Stratified Varietiesmentioning

confidence: 99%

K$K$‐Motives and Koszul duality

Eberhardt

2022

Bulletin of London Math Soc

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We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K$K$‐theory. For this, we introduce and study categories DKS(X)$\hbox{DK}_\mathcal {S}(X)$ of S$\mathcal {S}$‐constructible K$K$‐motivic sheaves on varieties X$X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S$\mathcal {S}$‐constructible mixed sheaves DSmix(X)$\hbox{D}^{mix}_\mathcal {S}(X)$ to DKS(X)$\hbox{DK}_\mathcal {S}(X)$. We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K$K$‐motivic sheaves on Langlands dual flag varieties.

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“…One can construct pointwise pure objects by different methods, for example, (1)using affinely stratified resolutions of singularities of closures of strata in

X

, see [18, Theorem 4.5]; (2)using contracting

\mathbb {G}_m

actions, see [33, Proposition 7.3. ]; (3)by an inductive process in the case of flag varieties, see [33, Lemma 6.6]. All of those methods can be used to show that all objects in

\hbox{D}^{mix}_\mathcal {S}(X)^{w=0}

and

\hbox{DK}_\mathcal {S}(X)^{w=0}

are pointwise pure in the case of flag varieties.…”

Section: Motivic Sheaves On Affinely Stratified Varietiesmentioning

confidence: 99%

K$K$‐Motives and Koszul duality

Eberhardt

2022

Bulletin of London Math Soc

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“…21 This is somewhat analogous to the theory of motives, where a motive has many different realisations, all of which are useful. In fact, recent work of Soergel and Wendt [SW18], Soergel, Wendt and Virk [SVW18] and Eberhardt and Kelly [EK19] shows that this is probably more than an analogy.…”

Section: The Many Faces Of the Hecke Categorymentioning

confidence: 99%

Modular representations and reflection subgroups

Williamson¹

2020

Preprint

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The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the "philosophy of deformations" both as a conceptual and computational tool, and suggest possible connections to Lusztig's "philosophy of generations". On the geometric side one can understand deformations in terms of localisation in equivariant cohomology. Recently Treumann and Leslie-Lonergan have added Smith theory, which provides a useful tool when considering mod p coefficients. In this context, we make contact with some remarkable work of Hazi. Using recent work of Abe on Soergel bimodules, we are able to reprove and generalise some of Hazi's results. Our aim is to convince the reader that the work of Hazi and Leslie-Lonergan can usefully be viewed as some kind of localisation to "good" reflection subgroups. These are notes for my lectures at the 2019 Current Developments in Mathematics at Harvard. Contents 1. Introduction 1 2. Philosophy of deformations 11 3. Affine reflection groups and Kazhdan-Lusztig theory 19 4. Localisation: equivariant, hyperbolic, Smith 33 5. The many faces of the Hecke category 46 6. The Hecke category and its localisations 54 7. Acknowledgements 69 References 69

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“…Remark 1.1. (1) The case of modular coefficients is work in progress joint with Shane Kelly and building on [EK19].…”

Section: Kosmentioning

confidence: 99%

K-Motives and Koszul Duality

Eberhardt¹

2019

Preprint

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We construct an ungraded version of Beilinson-Ginzburg-Soergel's Koszul duality, inspired by Beilinson's construction of rational motivic cohomology in terms of K-theory. For this, we introduce and study the category DK(X) of constructible K-motives on varieties X with an affine stratification. There is a natural and geometric functor from the category of mixed sheaves D mix (X) to DK(X). We show that when X is the flag variety, this functor is Koszul dual to the realisation functor from D mix (X ∨ ) to D(X ∨ ), the constructible derived category. Contents 1. Introduction 1 2. Motivic Sheaves après Cisinski-Déglise 3 3. Motives On Affinely Stratified Varieties 6 4. Flag Varieties and Koszul Duality 10 Appendix A. Weight Structures and t-Structures 13 References 16

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Mixed motives and geometric representation theory in equal characteristic

Cited by 10 publications

References 35 publications

K$K$‐Motives and Koszul duality

K$K$‐Motives and Koszul duality

Modular representations and reflection subgroups

K-Motives and Koszul Duality

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