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Representation Theory and Algebraic Geometry

A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg

Editors: Vladimir Baranovsky, Nicolas Guay, Travis Schedler

Publisher: Springer International Publishing

Book Series : Trends in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Table of Contents

About this book

The chapters in this volume explore the influence of the Russian school on the development of algebraic geometry and representation theory, particularly the pioneering work of two of its illustrious members, Alexander Beilinson and Victor Ginzburg, in celebration of their 60th birthdays. Based on the work of speakers and invited participants at the conference “Interactions Between Representation Theory and Algebraic Geometry”, held at the University of Chicago, August 21-25, 2017, this volume illustrates the impact of their research and how it has shaped the development of various branches of mathematics through the use of D-modules, the affine Grassmannian, symplectic algebraic geometry, and other topics. All authors have been deeply influenced by their ideas and present here cutting-edge developments on modern topics. Chapters are organized around three distinct themes:

Groups, algebras, categories, and representation theoryD-modules and perverse sheavesAnalogous varieties defined by quivers

Representation Theory and Algebraic Geometry will be an ideal resource for researchers who work in the area, particularly those interested in exploring the impact of the Russian school.

Frontmatter

Groups, Algebras, Categories, and Their Representation Theory

Frontmatter

On Semisimplification of Tensor Categories

Abstract

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S_n+p in characteristic p, where 0 ≤ n ≤ p − 1, and of the Deligne category \( \underline { \mathop {\mathrm {Rep}} \nolimits }^{\mathrm {ab}}S_t\), where t ∈ℕ. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of \(\mathfrak {sl}_2\). We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n), and PGL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.

Pavel Etingof, Victor Ostrik

Totally Aspherical Parameters for Cherednik Algebras

Abstract

We introduce the notion of a totally aspherical parameter for a rational Cherednik algebra. We get an explicit construction of the projective object defining the KZ functor for such parameters. We establish the existence of sufficiently many totally aspherical parameters for the groups G(ℓ, 1, n).

Ivan Losev

Microlocal Approach to Lusztig’s Symmetries

Abstract

We reformulate the De Concini-Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between Lusztig’s symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves.

Michael Finkelberg, Vadim Schechtman

D-Modules and Perverse Sheaves, Particularly on Flag Varieties and Their Generalizations

Frontmatter

Fourier-Sato Transform on Hyperplane Arrangements

Abstract

The theory of perverse sheaves can be said to provide an interpolation between homology and cohomology (or to mix them in a self-dual way). Since homology, sheaf-theoretically, can be understood as cohomology with compact support, interesting operations on perverse sheaves usually combine the functors of the types f_! and f_∗ or, dually, the functors of the types f^! and f^∗ in the classical formalism of Grothendieck.

Michael Finkelberg, Mikhail Kapranov, Vadim Schechtman

A Quasi-Coherent Description of the Category D -mod(GrGL(n))

Alexander Braverman, Michael Finkelberg

The Semi-infinite Intersection Cohomology Sheaf-II: The Ran Space Version

Abstract

This chapter is a sequel to [Ga1]. We define the semi-infinite category on the Ran version of the affine Grassmannian and study a particular object in it, denoted \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\), which we call the semi-infinite intersection cohomology sheaf.

Unlike the situation of [Ga1], this \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\) is defined as the middle of extension of the constant (more precisely, dualizing) sheaf on the basic stratum, in a certain t-structure. We give several explicit descriptions and characterizations of \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\): we describe its !- and *- stalks; we present it explicitly as a colimit; we relate it to the IC sheaf of Drinfeld’s relative compactification \(\overline {\operatorname {Bun}}_N\); we describe \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\) via the Drinfeld–Plucker formalism.

Dennis Gaitsgory

A Topological Approach to Soergel Theory

Abstract

We develop a “Soergel theory” for Bruhat-constructible perverse sheaves on the flag variety G∕B of a complex reductive group G, with coefficients in an arbitrary field

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. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a “multiplicative” coinvariant algebra and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of “Soergel modules” over this algebra. We also obtain a description of the derived category of unipotently T monodromic

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sheaves on G∕U (where U, T ⊂ B are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in

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, where

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is the

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-torus dual to T.

Roman Bezrukavnikov, Simon Riche

Varieties Associated to Quivers and Relations to Representation Theory and Symplectic Geometry

Frontmatter

Loop Grassmannians of Quivers and Affine Quantum Groups

Abstract

We construct for each choice of a quiver Q, a cohomology theory A, and a poset P a “loop Grassmannian” \({\mathcal G}^P(Q,A)\). This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a “dilation” torus \({\mathcal D}{ \subseteq } \mathbb {G}_m^2\) gives a quantization \({\mathcal G}^P_{\mathcal D}(Q,A)\). This construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program (Mirković, Some extensions of the notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić issue. No. 532, 53–74, 2017) and on the construction of affine quantum groups from generalized cohomology theories (Yang and Zhao, Quiver varieties and elliptic quantum groups, preprint. arxiv1708.01418).

Ivan Mirković, Yaping Yang, Gufang Zhao

Symplectic Resolutions for Multiplicative Quiver Varieties and Character Varieties for Punctured Surfaces

Abstract

We study the algebraic symplectic geometry of multiplicative quiver varieties, which are moduli spaces of representations of certain quiver algebras, introduced by Crawley-Boevey and Shaw in 2006, called multiplicative preprojective algebras. They are multiplicative analogues of Nakajima quiver varieties. They include character varieties of (open) Riemann surfaces fixing conjugacy class closures of the monodromies around punctures, when the quiver is “crab-shaped.” We prove that under suitable hypotheses on the dimension vector of the representations, or the conjugacy classes of monodromies in the character variety case, the normalizations of such moduli spaces are symplectic singularities and the existence of a symplectic resolution depends on a combinatorial condition on the quiver and the dimension vector. These results are analogous to those obtained by Bellamy and the first author in the ordinary quiver variety case and for character varieties of closed Riemann surfaces. At the end of the paper, we outline some conjectural generalizations to moduli spaces of objects in 2-Calabi–Yau categories.

Travis Schedler, Andrea Tirelli

Title: Representation Theory and Algebraic Geometry
Editors: Vladimir Baranovsky
Nicolas Guay
Travis Schedler
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-82007-7
Print ISBN: 978-3-030-82006-0
DOI: https://doi.org/10.1007/978-3-030-82007-7