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Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials

Simon Riche, Quan Situ

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in the principal block of the category of strongly $B$-equivariant $\mathfrak{g}$-modules (also called modular category $\mathcal{O}$), from a simple object to a costandard object, under the assumption that Lusztig's conjecture holds (which is known in large characteristic). The answer is given by a coefficient of a periodic Kazhdan--Lusztig polynomial associated with the corresponding affine Weyl group. Among other things, the proof uses a torus-equivariant version of the Koszul duality for $\mathfrak{g}$-modules constructed by the first author.

Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials

Abstract

Let be a connected reductive algebraic group over an algebraically closed field of positive characteristic, be its Lie algebra, and be a Borel subgroup. We prove a formula for the dimensions of extension groups, in the principal block of the category of strongly -equivariant -modules (also called modular category ), from a simple object to a costandard object, under the assumption that Lusztig's conjecture holds (which is known in large characteristic). The answer is given by a coefficient of a periodic Kazhdan--Lusztig polynomial associated with the corresponding affine Weyl group. Among other things, the proof uses a torus-equivariant version of the Koszul duality for -modules constructed by the first author.

Paper Structure

This paper contains 78 sections, 51 theorems, 380 equations.

Table of Contents

  1. Introduction
  2. Kazhdan--Lusztig combinatorics and representations of reductive grou-ps and their Lie algebras
  3. Statement
  4. Motivation
  5. Description of the proof
  6. Contents
  7. Acknowledgements
  8. Notation and conventions
  9. Grading shift
  10. Rings and modules
  11. Algebraic geometry
  12. Reductive groups
  13. Geometry associated with G
  14. (g,B)-modules
  15. Preliminaries
  16. ...and 63 more sections

Key Result

Theorem 1.1

Assume that $p>h$, and that Lusztig's conjecture holds. Then for any $y,w \in W_\mathrm{aff}$ we have

Theorems & Definitions (119)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof : Sketch of proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof : Sketch of proof
  • Remark 2.4
  • Remark 2.5
  • ...and 109 more