Preface
Contents
Part I Groups, Algebras, Categories, and Their Representation Theory
On Semisimplification of Tensor Categories
Contents
1 Introduction
2 Preliminaries
2.1 Tensor Ideals
2.2 Semisimplification of a Spherical Tensor Category
2.3 Generalization to Pivotal Karoubian Categories
3 General Results on Semisimplification of Tensor Categories
3.1 Splitting of the Semisimplification Functor for Tannakian Categories in Characteristic Zero, Reductive Envelopes, and the Jacobson-Morozov Lemma
3.2 Compatibility of Semisimplification with Equivariantization
3.3 Compatibility of Negligible Morphisms with Surjective Tensor Functors
4 Semisimplification of Representation Categories of Finite Groups in Characteristic p
4.1 The Result
4.2 Proof of Theorem 4.2
4.3 The Case of Sylow Subgroup*24pt of Prime Order
4.4 The Case of the Symmetric Group Sp+n, where n
4.5 Application: The Semisimplification of the Deligne Category Repab Sn
5 Semisimplification of Some Non-Symmetric Categories
5.1 Generic q
5.2 Roots of Unity
6 Surjective Symmetric Tensor Functors Between Verlinde Categories Verp(G)
7 Objects of Finite Type in Semisimplifications
8 Semisimplification of Tilt(GL(n)) when char(k)=2
Appendix A: Categorifications of Based Rings Attached to SO(3)
References
Totally Aspherical Parameters for Cherednik Algebras
Contents
1 Introduction
2 PKZ vs Harish-Chandra Module
2.1 Reminder on Cherednik Algebras
2.2 Reminder on Categories O
2.3 Totally Aspherical Parameters
2.4 Harish-Chandra Module vs PKZ
2.5 Total Asphericity vs Simplicity
2.6 Shifts
3 Cyclotomic Case
3.1 Type A
3.2 Quantized Quiver Varieties
3.3 Main Result in the Cyclotomic Case
3.4 Functors Ei,Fi
3.5 Absence of Finite Dimensional Representations
3.6 Proof of the Main Theorem
References
Microlocal Approach to Lusztig's Symmetries
Contents
1 Introduction
1.1 Coxeter Categories
1.2 Vanishing Cycles and Lusztig's Symmetries
1.3 Organization of the Paper
2 An Example
2.1 Algebra
2.3 Topology
2.7 Discussion
3 Coxeter Categories
3.1 Notations
3.2 The Fundamental Groupoid of hD'reg
3.3 The Fundamental Groupoid of NhD'/D''/hD'reg
3.7 Specialization
3.9 Comparison with the Appel-Toledano Laredo Coxeter Braided Tensor Categories
4 Algebra
4.1 Lusztig's Symmetries
4.4 A Coxeter Structure on RC, C
5 Topology
5.1 Erratum to bfs
5.2 A Review of bfs: Cohesive System and Algebra u-
5.3 A Review of bfs: Factorizable Sheaves
5.4 A Coxeter Structure on FS
5.7 Restriction Functors
5.8 Iterated Vanishing Cycles
6 Iterated Specialization and Microlocalization
6.1 Iterated Specialization
6.2.1 Cube
6.3 Iterated Microlocalization
6.5 Proof of Theorem 6.4
6.5.1 Subspace
6.5.2 Product
6.5.3 D-modules
6.5.4 D-modules on a Product Space
6.8 The End of the Proof
7 Discussion
7.1 Desiderata
7.2 Tilted Functors Φ
References
Part II D-Modules and Perverse Sheaves, Particularly on Flag Varieties and Their Generalizations
Fourier-Sato Transform on Hyperplane Arrangements
Contents
1 Introduction
1.1 Setup and Goals
1.2 Pattern of the Results
1.3 Structure of the Paper
2 Real and Complex Data Associated with Perverse Sheaves
2.1 The Real Setup
2.2 The Complex Setup
2.3 Real Data: Stalks and Hyperbolic Stalks
2.4 Hyperbolic Sheaves
3 Vanishing Cycles in Terms of Hyperbolic Sheaves
3.1 Background on Vanishing Cycles
3.2 The Complex Result
3.3 The Real Analog
3.4 Proof of Theorem 3.3
4 Specialization and Hyperbolic Sheaves
4.1 Generalities on Specialization
4.2 The Case of Sheaves on Arrangements
4.3 Specialization of Faces as a Continuous Map
4.4 The Real Result
4.5 Bispecialization
4.6 The Complex Result
5 Fourier Transform and Hyperbolic Sheaves
5.1 Generalities on the Fourier-Sato Transform
5.2 The Dual Arrangement
5.3 Big and Small Dual Cones
5.4 The Real Result
5.5 The Complex Result
6 Applications to Second Microlocalization
6.1 Microlocalization
6.2 Iterated Microlocalization
6.3 Bi-microlocalization
6.4 Comparisons in the Linear Case
6.5 Proof of Theorem 6.7
6.6 Proof of Theorem 6.6
References
A Quasi-Coherent Description of the Category D-mod(GrGL(n))
Contents
1 Introduction and Statement of the Results
1.1 General Notation
1.2 The Main Conjecture: GL(n)-case
1.4 The Main Conjecture: GL(2)-case
1.7 Fiberwise Version
2 Proof of Theorem 1.8(1)
2.1 Sketch of the Proof
2.2 The Map ιL
2.3 Proof of (i)
2.4 Proof of (ii)
2.5 Proof of (iii)
3 Proof of Theorem 1.8(2)
3.1 Reduction to SL(2)
3.2 Koszul Duality
3.4 Equivariant Cohomology
3.7 The Functor
3.8 Computing Ext's
3.11 The End of the Proof
3.12 Abelian Equivalence
4 Proof of Theorem 1.8(3)
4.1 Compact Objects in D-modH(X)
4.3 The Cohomology Functor
4.5 Compact Objects in DO(Gr)
4.6 End of the Proof
References
The Semi-infinite Intersection Cohomology Sheaf-II: The Ran Space Version
Contents
1 Introduction
1.1 What Are Trying to Do?
1.1.1
1.1.2
1.1.4
1.1.5
1.1.6
1.2 What Is Done in This Paper?
1.2.1
1.2.2
1.2.3
1.2.4
1.3 Organization
1.3.1
1.3.2
1.3.3
1.3.4
1.3.5
1.4 Background, Conventions, and Notation
1.4.1
1.4.2
1.4.3
1.4.4
1.4.5
1.4.6 Sheaves on Prestacks
1.4.7
1.4.8
2 The Ran Version of the Semi-infinite Category
2.1 The Ran Grassmannian
2.1.1
2.1.2
2.1.3
2.1.4
2.2 The Semi-infinite Category
2.2.1
2.2.2
2.2.3
2.3 Stratification
2.3.1
2.3.2
2.3.5
2.3.6
2.3.7
2.3.8
2.3.10
2.4 The Category on a Single Stratum
2.4.1
2.4.3 Proof of Proposition 2.4.2
2.5 Interaction Between the Strata
2.5.1
2.5.2
2.5.3
2.5.7
2.6 An Aside: The ULA Property
2.6.1
2.6.2
2.7 An Application of Braden's Theorem
2.7.1
2.7.3
2.7.4
3 The t-Structure and the Semi-infinite IC Sheaf
3.1 The t-Structure on the Semi-infinite Category
3.1.1
3.1.3
3.1.5
3.1.6
3.1.8
3.2 Definition of the Semi-infinite IC Sheaf
3.2.1
3.2.3
3.2.4
3.3 Digression: From Commutative Algebras to Factorization Algebras
3.3.1
3.3.2
3.3.3
3.3.4 An Example
3.3.5
3.4 Restriction of IC∞2Ran to Strata
3.4.1
3.4.2
3.4.4
3.5 Digression: Categories over the Ran Space
3.5.1
3.5.2
3.5.3
3.5.4
3.5.5
3.5.6
3.5.7
3.5.8
3.5.9
3.5.10
3.6 Presentation of IC∞2 as a Colimit
3.6.1
3.6.2
3.6.3
3.6.4
3.6.5
3.6.7
3.6.8
3.7 Presentation of IC∞2Ran as a Colimit
3.7.1
3.7.6
3.7.8
3.8 Description of the *-Restriction to Strata
3.8.1
3.8.2
3.8.3
3.8.4
3.8.5
3.8.6
3.9 Proof of Coconnectivity
3.9.1
3.9.2
3.9.4
3.9.5
4 The Semi-infinite IC Sheaf and Drinfeld's Compactification
4.1 Drinfeld's Compactification
4.1.1
4.1.2
4.1.3
4.2 The Global Semi-infinite Category
4.2.1
4.2.2
4.2.3
4.2.4
4.2.6
4.2.7
4.3 Local vs. Global Compatibility for the Semi-infinite IC Sheaf
4.3.1
4.3.2
4.3.4
4.4 The Local vs. Global Compatibility for the Semi-infinite Category
4.4.1
4.4.3
4.4.7
4.5 Proof of Proposition 4.4.5
4.5.1
4.5.2
4.5.3
4.6 The Key Isomorphism
4.6.1
4.6.2
4.6.8
4.6.10
4.7 Proof of Proposition 4.6.3
4.7.1
4.7.2
4.7.3
4.7.4
4.7.5
4.7.6
4.7.7
4.7.8
4.7.9
4.8 Relation to the IC Sheaf on Zastava Spaces
4.8.1
4.8.2
4.8.4 Proof of Proposition 4.8.3
4.9 Computation of Fibers
4.9.2
4.9.4
5 Unital Structure and Factorization
5.1 Unital Structure on the Affine Grassmannian
5.1.1
5.1.4
5.1.5
5.1.6
5.1.8
5.2 Unital Structure on the Strata
5.2.1
5.2.4
5.2.6
5.3 Local-to-Global Comparison, Revisited
5.3.1
5.3.3
5.3.4
5.4 The t-Structure on the Unital Category
5.4.1
5.4.2
5.4.4
5.5 Comparison with IC on Zastava Spaces, Continued
5.5.1
5.5.2
5.5.3
5.6 Factorization Structure on IC∞2
5.6.1
5.6.2
5.6.3
5.6.4
5.6.5
5.6.8
5.7 Factorization and Zastava Spaces
5.7.1
5.7.2
5.7.3
5.7.4
6 The Hecke Property of the Semi-infinite IC Sheaf
6.1 Pointwise Hecke Property
6.1.1
6.1.2
6.1.3
6.1.4
6.1.5
6.1.6
6.1.7
6.2 Categories over the Ran Space, Continued
6.2.1
6.2.2
6.2.3
6.2.4 Examples
6.3 Digression: Right-Lax Central Structures
6.3.1
6.3.2
6.3.3
6.3.5
6.3.7
6.3.8
6.3.9
6.4 Hecke and Drinfeld–Plücker Structures
6.4.1
6.4.2
6.4.3
6.4.4
6.4.5
6.4.6
6.4.8
6.5 The Hecke Property-Enhanced Statement
6.5.1
6.5.2
6.5.3
6.5.4
6.5.6
6.6 Recovering the Pointwise Hecke Structure
6.6.1
6.6.2
7 Local vs. Global Compatibility of the Hecke Structure
7.1 The Relative Version of the Ran Grassmannian
7.1.1
7.1.2
7.1.3
7.2 Hecke Property in the Global Setting
7.2.1
7.2.2
7.2.3
7.3 Local vs. Global Compatibility
7.3.1
7.3.3
7.3.4
7.4 Proof of Theorem 7.3.5
7.4.1
7.4.2
7.4.3
7.4.4
Appendix A: Proof of Theorem 4.4.4
A.1 The Space of G-Bundles with a Generic Reduction
A.1.1
A.1.2
A.1.4
A.1.6
A.1.7
A.1.8
A.1.9
A.1.11
A.2 Toward the Proof of Theorem A.1.10
A.2.1
A.2.4
A.2.6 Interlude: The Relative Ran Space
A.2.8 Proof of Lemma 2.3.3 for X Proper
A.2.9 Proof of Proposition A.2.2
A.3 Proof of Theorem A.2.3
A.3.1
A.3.4 Proof of Theorem A.3.2
A.3.5
A.3.7
A.3.8
A.4 Proof of Theorem A.3.3 for H Reductive
A.4.1
A.4.2
A.4.3
A.4.4
References
A Topological Approach to Soergel Theory
Contents
1 Introduction
1.1 Soergel Theory
1.2 Geometric Version
1.3 Monodromy
1.4 Free-Monodromic Deformation
1.5 Identification of End(Tw0)
1.6 The Functor V
1.7 Some Remarks
1.8 Contents
2 Monodromy
2.1 Construction
2.2 Basic Properties
2.3 Monodromy and Equivariance
3 Completed Category
3.1 Definition
3.2 The Free-Monodromic Local System
3.3 ``Averaging'' with the Free-Monodromic Local System
4 The Case of the Trivial Torsor
4.1 Description of D(A/A) in Terms of Pro-complexes of RA-Modules
4.2 Some Results on Pro-complexes of RA-Modules
4.3 Description of D(A/A) in Terms of Complexes of RA-Modules
5 The Perverse t-Structure
5.1 Recollement
5.2 Definition of the Perverse t-Structure
5.3 Standard and Costandard Perverse Sheaves
5.4 Tilting Perverse Sheaves
5.5 Classification of Tilting Perverse Sheaves
6 Study of Tilting Perverse Objects
6.1 Notation
6.2 Right and Left Monodromy
6.3 The Associated Graded Functor
6.4 Monodromy and Coinvariants
6.5 The Case of Ts
6.6 Properties of Tw0
7 Convolution
7.1 Definition
7.2 Convolution and Monodromy
7.3 Extension to the Completed Category
7.4 Convolution of Standard, Costandard, and Tilting Objects
8 Variations on Some Results of Kostant–Kumar
8.1 The Pittie–Steinberg Theorem
8.2 Some RT-Modules
8.3 An Isomorphism of RT-Modules
8.4 A Different Description of the Algebra RT
9 Endomorphismensatz
9.1 Statement and Strategy of Proof
9.2 A Special Case
9.3 The General Case
10 Variant: The étale Setting
10.1 Completed Derived Categories
10.2 Soergel's Endomorphismensatz
10.3 Whittaker Derived Category
10.4 Geometric Construction of Tw0
11 Soergel Theory
11.1 The Functor V
11.2 Image of Ts
11.3 Monoidal Structure: étale Setting
11.4 Monoidal Structure: Classical Setting
11.5 Soergel Theory
12 Erratum to ab
References
Part III Varieties Associated to Quivers and Relations to Representation Theory and Symplectic Geometry
Loop Grassmannians of Quivers and Affine Quantum Groups
Contents
1 Introduction
1.1 Loop Grassmannians Associated to Quivers
1.2 Construction
1.3 ``Quantum Nature'' of GPD(Q,A)
1.4 Contents
2 Recollections on Cohomology Theories
2.1 Equivariant-Oriented Cohomology Theories
2.2 Thom Line Bundles
2.2.1 ΘG(V) for a Representation V of G
2.2.2 Thom Line Bundles Θ(f) of Maps f
3 Loop Grassmannians and Local Spaces
3.1 Loop Grassmannians
3.1.1 Global (Beilinson-Drinfeld) Loop Grassmannians G HC (G)
3.1.2 The Abel-Jacobi Map
3.2 The T-Fixed Points in Semi-infinite Varieties Sα
3.2.1 Tori
3.2.2 The ``Semi-Infinite'' Orbits Sλ
3.2.3 The T-Fixed Points
3.2.4 The Kamnitzer-Knutson Program of Reconstructing MV Cycles
3.3 Local Spaces Over a Curve
3.3.1 Local Spaces
3.3.2 Classification of Local Line Bundles on the Colored Hilbert Scheme
3.4 A Generalization GP(I,Q) of Loop Grassmannians of Reductive Groups
3.4.1 Local Line Bundles from Loop Grassmannians
3.4.2 Zastava Spaces of Local Line Bundles
3.4.3 Weak Flatness of Zastavas
3.4.4 Grassmannians from Based Symmetric Bilinear Forms
3.4.5 Zastava Spaces Zα(G) for Groups
3.4.6 Reconstruction of Loop Grassmannians G(G) Associated to Groups
3.4.7 The ``Homological'' Aspect of GP(I,Q)
4 Local Line Bundles from Quivers
4.1 Quivers
4.1.1 Dilation Torus D
4.1.2 The Extension Correspondence for Quivers
4.2 Thom Line Bundles
4.2.1 Classical Thom Bundles for Quivers
4.3 Cotangent Versions of the Extension Diagram
4.3.1 The (Co)tangent Functoriality
4.3.2 Stacks
4.4 The A-Cohomology of the Cotangent Correspondence for Extensions
4.4.1 Connected Components of the Cotangent Correspondence
4.4.2 Line Bundles from the Cotangent Correspondence
4.4.3 Dilations
4.5 D-Quantization of the Monoid (HG I,+)
4.5.1 Local and Biextension Line Bundles LD(Q,A) and LD(Q,A)
4.5.2 Convolutions and Biextensions
4.5.3 Quantum Groups U+D(Q,A)UD(Q,A)
4.5.4 Local Line Bundle from Zastava and U+D(Q,A)
5 Loop Grassmannians GPD(Q,A) and Quantum Locality
5.1 The ``Classical'' Loop Grassmannians GP(Q,A)
5.2 Quantization Shifts Diagonals
5.2.1 Deformation of a Thom Divisor from an Additional Torus D
5.2.2 Quantum Diagonals
5.3 Quantum Locality
5.3.1 m-Locality
5.3.2 Some Expectations
Appendix 1: Loop Grassmannians with a Condition
Moduli of Finitely Supported Maps
The Subfunctor G(G,Y)G(G) Given by ``Condition Y''
The Closure of S0
Proof of the Proposition 3.2.3
Appendix 2: Calculation of Thom Line Bundles from YZell
References
Symplectic Resolutions for Multiplicative Quiver Varieties and Character Varieties for Punctured Surfaces
Contents
1 Introduction
1.1 Motivation
1.2 Summary of Results on Character Varieties
1.3 Multiplicative Quiver Varieties with Special Dimension Vectors
1.4 Character Varieties as (Open Subsets of) Multiplicative Quiver Varieties
1.5 General Dimension Vectors
1.6 Outline of the Paper
2 Multiplicative Quiver Varieties
2.1 Preliminaries on Quivers and Root Systems
2.2 Multiplicative Preprojective Algebras
2.3 Reflection Functors for Λq(Q)
2.4 Moduli of Representations of Λq(Q)
2.5 Reflection Isomorphisms
2.6 Poisson Structure on Mq, θ(Q, α)
2.7 Stratification by Representation Type
2.8 The Set Σq,θ
3 Punctured Character Varieties as Multiplicative Quiver Varieties
4 Singularities of Multiplicative Quiver Varieties
4.1 Singular Locus of Mq, θ(Q, α) for αΣq,θ
4.2 Generalities on Symplectic Singularities
4.3 The q-Indivisible Case
4.4 The q-Divisible Case
4.5 Proof of Corollary 1.6
4.6 The Anisotropic Imaginary (p(α),n)= (2,2) Case
5 Combinatorics of Multiplicative Quiver Varieties
5.1 (2,2) Cases for Crab-Shaped Quivers
6 General Dimension Vectors and Decomposition
6.1 Flat Roots
6.2 Fundamental and Flat Roots Not in Σq,θ
6.3 Canonical Decompositions
6.4 Symplectic Resolutions for q-Indivisible Flat Roots
6.5 Symplectic Resolutions for General α
6.6 Classifications of Symplectic Resolutions of Punctured Character Varieties
6.7 Proof of Theorems 1.1 and 1.3
7 Open Questions and Future Directions
7.1 Non-emptiness of Multiplicative Quiver Varieties
7.2 Refined Decompositions for Multiplicative Quiver Varieties
7.3 Symplectic Resolutions and Singularities
7.4 Moduli of Parabolic Higgs Bundles and the Isosingularity Theorem
7.5 Moduli Spaces in 2-Calabi–Yau Categories
7.6 Character Varieties and Higgs Bundles for Arbitrary Groups
References
