Alphabeta MathThe Kazhdan–Lusztig Character and Multiplicity Theorem
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1 · Prerequisites
Abstract Algebra
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2 · Logical flowchart
3 · Items
Definition
The regular integral dot-orbit labelling convention
Lemma
Projective Verma multiplicities equal Soergel standard ranks
Theorem
Projective Verma multiplicities are Kazhdan–Lusztig values at one
Theorem
Simple Verma composition multiplicities are Kazhdan–Lusztig values
Theorem
The Kazhdan–Lusztig character formula in regular integral category O
Proposition
Translation identifies the regular integral blocks
Remark
The full Ext version requires the Koszul grading of category O
Sources
- Wolfgang Soergel, Kategorie O, perverse Garben und Moduln ueber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990) 421-445 (MPIM preprint 1989/46, complete German text)
- Lin Chen, lecture notes (Spring 2024), Lecture 8: Hecke algebras and the Kazhdan-Lusztig conjecture (complete 10-page lecture)
- Lin Chen, lecture notes (Spring 2024), Lecture 9: standard filtrations, BGG reciprocity and translation functors (complete 7-page lecture)
- Anders Bjoerner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer (author-hosted complete 370-page PDF)