Alphabeta MathAffine Hecke Algebras Beyond Type A
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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
- Characters and the Orthogonality Relations
- 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 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
- Induced Representations, Frobenius Reciprocity and Applications
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- 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
- 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
Algebraic Topology
Categorification and Geometric Representation Theory
Category Theory
Combinatorics
Commutative Algebra
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
- Crystallographic Root Lattices and Weyl Group Interfaces
- Finite Coxeter Diagrams and Complete Classification
- Permutation Statistics, Inversions and Eulerian Numbers
- Real Forms and Reflection Geometry
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
Functional Analysis
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Complex Subspace Realification
- Finite Dimensional Normed Spaces and Riesz Lemma
- Hilbert Space Geometry and Riesz Representation
- Locally Convex Spaces and Continuous Separation
- Normed and Banach Spaces
- The Analytic Hahn Banach Theorem
Group Theory
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
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- 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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convex and Semicontinuous Functions on Rⁿ
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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 Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Representation Theory of Groups
Scheme Theory
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
Symmetric Groups and Symmetric Functions
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
The Iwahori–Matsumoto affine Hecke algebra of a based root datum
Lemma
A real affine-Hecke parameter greater than one has infinite quantum characteristic
Lemma
The lattice tree and Iwahori orbits for SL2(Q3)
Lemma
The extended Iwahori–Matsumoto basis
Lemma
Canonical affine geometry and wall parameters of a root datum
Definition
The formal Bernstein presentation for a reduced based root datum
Lemma
Signed alcove products and the oriented affine Hecke basis
Theorem
The SL2(Q3) Iwahori convolution corner has affine parameters 3
Lemma
The Bernstein cross-relation quotients are Laurent polynomials
Theorem
General Bernstein PBW and the Iwahori–Matsumoto comparison
Definition
Parabolic affine Hecke algebras and their weight characters
Remark
The general Bernstein PBW construction and source boundary
Definition
Weight temperedness and positive Langlands data
Lemma
Parabolic induction, central orbit blocks and weight filtrations
Lemma
Dominant projection and strict parabolic weight decrease
Theorem
The unique irreducible quotient of an affine Hecke standard module
Theorem
The tempered Langlands classification for affine Hecke algebras
Remark
The local tempered Langlands proof and the selected-group boundary
Sources
- Maarten Solleveld, Affine Hecke Algebras and Their Representations
- Dan Ciubotaru, Tutorial on Affine Hecke Algebras
- Maarten Solleveld, Affine Hecke algebras and their representations, §1.2
- Maarten Solleveld, Affine Hecke algebras and their representations, §§1.3–1.4
- Ulrich Görtz, Alcove walks and nearby cycles on affine flag manifolds
- Maarten Solleveld, On the classification of irreducible representations of affine Hecke algebras with unequal parameters