Alphabeta MathBeilinson–Bernstein Localization and Category O
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Finite Fields and Cyclotomic Extensions
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Semidirect Products, Automorphism Groups and Split Extensions
- Simple Field Extensions and the Construction of the Complex Numbers
- Solvability by Radicals and Kummer Theory
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Classical Complex Algebraic Actions and Affine Embeddings
- Dimension Constructible Images and Dimensions of Fibres
- Groups of Multiplicative Type and Arithmetic Tori
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
- Normal Varieties, Normalization, and Zariski's Main Theorem
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Projective GIT from Linearized Line Bundles
- Smooth-Projective Serre Duality and Flag-Variety Line Bundles
- Split Reductive Root Systems, Bruhat Cells, and Parabolics
- Unipotent and Solvable Groups and Borel Fixed Points
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Algebraic Topology
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
- Algebraic Zariski Main for Quasi-Finite Morphisms
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Homogeneous Resultants and Projective Intersection Length
- Integral Extensions and Going Up
- Inverse Limits and Noetherian Completion
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Complex Analysis
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Coxeter Groups
- Affine Reflections, Coroot Translations, and Alcoves
- Canonical Roots, Signs, and Faithful Reflections
- Conjugacy in Sₙ, Generation, and the Simplicity of Aₙ
- Coxeter Presentations, Exchange, and Reduced Word Theorems
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- Finite Coxeter Diagrams and Complete Classification
- Real Forms and Reflection Geometry
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
Differential Geometry
- Distributions Integral Manifolds and the Frobenius Theorem
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Functional Analysis
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Homological Algebra
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- Delta Functors and Universality
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- Derived Functors
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- Ext and Balanced Resolutions
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- Projective and Injective Resolutions
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- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Lie Theory
- Borel Weil and Borel Weil Bott
- Cartan Subalgebras and Root Space Decompositions
- Category O Finiteness Duality and Blocks
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Harish Chandra Isomorphism Casimir and Central Characters
- Highest Weight Theory for Complex Semisimple Lie Algebras
- Lie Algebra Representations, Enveloping Algebras, and PBW
- Lie Groups, Invariant Fields, and the Exponential Map
- Lie Subgroups, Actions, and Homogeneous Spaces
- Projectives Standard Filtrations and Bgg Reciprocity
- Root Systems, Dynkin Diagrams, and the Cartan-Killing Classification
- Semisimple Lie Algebras, Cohomology, and Levi Theory
- Solvable and Nilpotent Lie Algebras
- Verma Modules and Shapovalov Forms
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
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- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Representation Theory of Groups
Scheme Theory
- Affine Group Schemes, Hopf Algebras, and Rational Representations
- Affine Schemes and the Structure Sheaf
- Algebraic Group Actions, Orbits, Stabilizers, and Controlled Quotients
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Group Schemes of Finite Type over a Field
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Lie Algebras and Infinitesimal Group Schemes
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Applications of the Fundamental Group
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Definition
Flag-variety twisted differential operators and their modules
Lemma
Complex-weight separation of Harish-Chandra characters
Lemma
Global sections on module sheaves have ordinary sheaf cohomology as derived functors
Lemma
Cech cohomology of line bundles on the projective line
Lemma
The principal nilpotent Borel orbit by height elimination
Lemma
Quasicoherent modules are filtered unions of coherent subsheaves and cohomology commutes with these colimits
Lemma
PBW and integral line-bundle twisting for flag TDOs
Lemma
Normality of the nilpotent cone and Springer functions
Lemma
Partial-flag twist and local product decomposition
Theorem
Global sections of twisted differential operators on the flag variety
Definition
Global sections and localization at a fixed infinitesimal character
Lemma
Ample translation maps split by central-character separation
Lemma
Localization is left adjoint to global sections
Theorem
Antidominant twists are D-affine and regular ones are conservative
Lemma
The unit V → Γ Loc_κ(V) is an isomorphism for antidominant twists
Lemma
The singular simple-parabolic kernel on the flag variety
Theorem
The singular antidominant Serre quotient
Theorem
Regular antidominant Beilinson–Bernstein localization
Corollary
The integral regular localization equivalence restricts to Category O
Theorem
Kashiwara's singular parabolic kernel filtration
Lemma
Relative rank-one classification of the singular differential kernel
Lemma
Exact twisted associated bundles and simple-parabolic descent
Lemma
A fixed singular Verma chain with globally generated final bundle
Sources
- Masaki Kashiwara, Representation theory and D-modules on flag varieties, Astérisque 173–174 (1989)
- Yi Sun, D-modules and representation theory
- Pavel Etingof, Representations of Lie Groups, MIT 18.757 (2023)
- The Stacks Project, Cohomology of Sheaves, Lemma 20.8.1
- The Stacks Project, Sheaves of Modules, Section 19.5
- The Stacks Project, Properties of Schemes, Lemma 28.23.3
- The Stacks Project, Cohomology of Schemes, Lemma 30.9.1
- Ravi Vakil, The Rising Sea (29 August 2022)