Alphabeta MathSingular and Parabolic Kazhdan–Lusztig Theory
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1 · Prerequisites
Abstract Algebra
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Set Theory
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Topology
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- The Fundamental Group
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- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Definition
The parabolic Hecke module and the antispherical standard basis
Lemma
Standard bases, right actions and bar involutions on the parabolic Hecke modules
Theorem
Existence and uniqueness of the parabolic Kazhdan–Lusztig bases
Definition
The parabolic Kazhdan–Lusztig bases and polynomials
Theorem
Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones
Definition
The singular integral block and its dot stabilizer
Lemma
The facet tensor-weight exclusion lemma
Theorem
Translation to the wall on Verma, simple and projective objects
Theorem
Singular category O multiplicities are parabolic Kazhdan–Lusztig values
Definition
Standard parabolic and Levi subalgebras from a subset of simple roots
Definition
Parabolic category O for a standard parabolic
Theorem
Parabolic category O is a highest weight category
Theorem
Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values
Theorem
The parabolic Kazhdan–Lusztig character formula
Sources
- Nicolas Libedinsky and Geordie Williamson, The anti-spherical category (author-hosted complete PDF of the article)
- Volodymyr Mazorchuk and Catharina Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, Trans. Amer. Math. Soc. 357 (2005) 2939-2973 (author-hosted published version)
- Anders Bjoerner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer 2005 (author-hosted complete 370-page PDF)
- James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, AMS Graduate Studies in Mathematics 94
- Lin Chen, lecture notes (Spring 2024), Lecture 9: standard filtrations, BGG reciprocity and translation functors (complete 7-page lecture)
- Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrden, Some homological properties of category O VI, Documenta Mathematica 26 (2021) 1237-1269 (complete article)
- Wolfgang Soergel, Character formulas for tilting modules over Kac-Moody algebras (22-page author preprint)
- Volodymyr Mazorchuk, Lectures on Algebraic Categorification
- Anne Moreau, M2 Representation Theory 2025–2026 (current complete 129-page author notes)