Representation Theory and Algebraic Geometry
A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg
- 2022
- Book
- Editors
- Vladimir Baranovsky
- Nicolas Guay
- Travis Schedler
- Book Series
- Trends in Mathematics
- Publisher
- Springer International Publishing
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.
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Table of Contents
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Frontmatter
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Groups, Algebras, Categories, and Their Representation Theory
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Frontmatter
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On Semisimplification of Tensor Categories
Pavel Etingof, Victor OstrikThe chapter delves into the semisimplification of tensor categories, a process that simplifies complex algebraic structures by eliminating negligible morphisms. It begins with an introduction to the concept, tracing its origins to algebraic geometry and the theory of motives. The text then explores general results about semisimplification, including its compatibility with equivariantization and surjective tensor functors. Notably, it applies classical results from modular representation theory to show that the semisimplification of the category of representations of a finite group in characteristic p is naturally equivalent to that of its Sylow p-subgroup's normalizer. The chapter also highlights practical applications, such as the computation of semisimplifications in specific examples and the existence of reductive envelopes for certain algebraic groups. These insights provide a deeper understanding of the structure and behavior of tensor categories, making the chapter a valuable resource for specialists in the field.AI Generated
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AbstractWe 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 Sn+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. -
Totally Aspherical Parameters for Cherednik Algebras
Ivan LosevThe chapter delves into the concept of totally aspherical parameters for Cherednik algebras, focusing on their definition and implications. It explores the triangular decomposition of the rational Cherednik algebra H c ( W ) and the properties of its spherical subalgebra eH c ( W ) e. The text introduces the notion of totally aspherical parameters and their equivalence to the simplicity of the algebra eH c ( W ) e. It also investigates the relationship between the Harish-Chandra module and the projective object P KZ, highlighting the criteria for their equivalence. Additionally, the chapter applies these concepts to specific cases, such as the symmetric group and the groups G ( ℓ, 1, n ), providing insights into the existence and properties of totally aspherical parameters in these contexts.AI Generated
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AbstractWe 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). -
Microlocal Approach to Lusztig’s Symmetries
Michael Finkelberg, Vadim SchechtmanThe chapter 'Microlocal Approach to Lusztig’s Symmetries' delves into the intricate relationship between Lusztig’s symmetries and Coxeter categories. It introduces a topological reformulation of Coxeter categories, which provides a fresh perspective on the subject. The authors explore the vanishing cycles and Lusztig’s symmetries, presenting conjectures and detailed analyses. The chapter also includes an example in type A2, comparing the action of Lusztig’s symmetries in weight spaces with the monodromy action in vanishing cycles of perverse sheaves. The authors propose a Coxeter structure on the category of factorizable sheaves and conjecture that this structure is equivalent to the Coxeter structure on the category of integrable modules over Lusztig’s small quantum group. The chapter concludes with a discussion of iterated specialization and microlocalization, highlighting the importance of these techniques in the study of Lusztig’s symmetries.AI Generated
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AbstractWe 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.
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D-Modules and Perverse Sheaves, Particularly on Flag Varieties and Their Generalizations
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Frontmatter
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Fourier-Sato Transform on Hyperplane Arrangements
Michael Finkelberg, Mikhail Kapranov, Vadim SchechtmanThe chapter delves into the intricate relationship between homology and cohomology through the lens of perverse sheaves, specifically focusing on the Fourier-Sato Transform on hyperplane arrangements. It introduces the concept of hyperbolic sheaves, which uniquely determine perverse sheaves through generalization and specialization maps. The text describes the effect of operations such as forming vanishing cycles, specialization, and the Fourier-Sato transform directly in terms of hyperbolic sheaves. It also highlights the importance of these operations in understanding the weight components of highest weight modules over quantized Kac-Moody algebras. Additionally, the chapter explores the microlocalization along linear subspaces and compares different definitions of second microlocalization in the linear case, providing a thorough and insightful analysis of these complex mathematical concepts.AI Generated
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AbstractThe 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. -
A Quasi-Coherent Description of the Category D -mod(GrGL(n))
Alexander Braverman, Michael FinkelbergThe chapter delves into the derived category of quasi-coherent sheaves and D-modules on derived stacks, establishing general notation and definitions. It presents the main conjecture for the case of local systems and morphisms pulled back from specific stacks. The proofs for specific cases are meticulously provided, focusing on the intersection of orbits and the behavior of D-modules on these orbits. The chapter highlights the conditions under which these orbits support equivariant D-modules and computes the convolution of specific sheaves, demonstrating the intricate relationships within the category.AI Generated
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The Semi-infinite Intersection Cohomology Sheaf-II: The Ran Space Version
Dennis GaitsgoryThe chapter delves into the construction of a variant of the semi-infinite intersection cohomology sheaf on the Ran space, denoted as . This variant is closely related to the original construction but is given by the procedure of middle extension in a certain t-structure. The Ran space, attached to a smooth curve X, is utilized to achieve a more intrinsic characterization of the sheaf. The construction retains the relation to the original sheaf and is shown to be invariant under the operation of 'throwing in' more points without altering the G-bundle. The chapter also explores the unitality and eigen-property of the sheaf with respect to Hecke functors for G and T, and establishes the compatibility of these properties with those of the original sheaf. Additionally, the text provides an explicit presentation of the sheaf as a colimit and describes its *- and !-restrictions to the strata. The chapter concludes by relating the Ran version of the semi-infinite IC sheaf to the intersection cohomology sheaf of Drinfeld's compactification, demonstrating its significance in the broader context of geometric Langlands theory.AI Generated
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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.
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A Topological Approach to Soergel Theory
Roman Bezrukavnikov, Simon RicheThe chapter 'A Topological Approach to Soergel Theory' delves into the mathematical framework of Soergel theory, which studies the principal block of the Bernstein–Gelfand–Gelfand category of a complex semisimple Lie algebra. The author begins by introducing Soergel theory and its key results, the Endomorphismensatz and Struktursatz, which describe the category of projective objects in terms of commutative algebra. The chapter then transitions to a geometric approach, valid for arbitrary field coefficients, and explores the monodromy of perverse sheaves on the Langlands dual flag variety. Additionally, the author discusses the free-monodromic deformation and its implications for the essential image of certain functors. The chapter concludes with a description of the monoidal structure on the category of tilting objects, highlighting the strong connection between Soergel theory and geometric constructions.AI Generated
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AbstractWe 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
. 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 
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 
, where 
is the 
-torus dual to T.
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Varieties Associated to Quivers and Relations to Representation Theory and Symplectic Geometry
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Frontmatter
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Loop Grassmannians of Quivers and Affine Quantum Groups
Ivan Mirković, Yaping Yang, Gufang ZhaoThe chapter 'Loop Grassmannians of Quivers and Affine Quantum Groups' delves into the mathematical constructs of loop Grassmannians and affine quantum groups, emphasizing their relevance to the work of influential mathematicians Sasha Beilinson and Vitya Ginzburg. It introduces the concept of loop Grassmannians associated with quivers, which provides a general framework for studying representations of algebraic groups and their loop groups. The author reconstructs loop Grassmannians from quivers using cohomology theories, leading to a deeper understanding of their geometric and algebraic properties. The text also explores the quantum generalization of these structures, offering a novel perspective on the interplay between geometry and quantum mechanics. The chapter is rich in technical details and theoretical insights, making it a valuable resource for researchers and mathematicians interested in algebraic geometry and representation theory.AI Generated
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AbstractWe 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). -
Symplectic Resolutions for Multiplicative Quiver Varieties and Character Varieties for Punctured Surfaces
Travis Schedler, Andrea TirelliThis chapter delves into the symplectic algebraic geometry of coarse moduli spaces of multiplicative preprojective algebras, which are multiplicative analogues of Nakajima quiver varieties. It tackles two main problems: determining whether these varieties are symplectic singularities and classifying cases where they admit symplectic resolutions. The study builds on previous work by Crawley-Boevey and Shaw, applying Drezet’s factoriality criteria and Flenner’s theorem. Additionally, the chapter explores character varieties for punctured surfaces, providing a detailed analysis of their symplectic properties and offering a summary of results on these varieties. This work is significant as it contributes to the growing theory of symplectic duality and quantum cohomology, with potential applications in representation theory and mathematical physics.AI Generated
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AbstractWe 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.
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- Title
- Representation Theory and Algebraic Geometry
- Editors
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Vladimir Baranovsky
Nicolas Guay
Travis Schedler
- Copyright Year
- 2022
- 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
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