Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tilting modules for reductive algebraic groups: characters and support varieties

Pramod N. Achar, Simon Riche

TL;DR

The notes address the problem of understanding characters and support varieties of tilting modules for reductive groups over fields of positive characteristic. The authors develop and apply a $p$-canonical framework, connecting representation-theoretic questions to the geometry of the nilpotent cone via the Springer resolution and the Lusztig–Vogan bijection, with a focus on tilting modules as a bridge to simple-character information. They prove a tilting-character formula in terms of antispherical $p$-Kazhdan–Lusztig polynomials, and establish large-p results that align with Andersen–Jantzen–Soergel style predictions, while also pursuing more geometric approaches through the Finkelberg–Mirković conjecture and categorical structures. The work then extends to support varieties, proving Humphreys-type conjectures in significant cases, and articulates a rich program linking cohomology of Frobenius kernels, perverse-coherent sheaves, and tensor-ideal structures, with a coherent Lusztig–Vogan framework across characteristics. Together, these results illuminate the deep interplay between modular representation theory, geometric representation theory, and categorical methods, offering both concrete formulas and a broad geometric paradigm for future progress.

Abstract

These notes are our contribution to the Proceedings of the ICM 2026. We discuss some results we have obtained (in part jointly with coauthors) regarding the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. These statements mainly concern tilting modules, in particular their characters and support varieties.

Tilting modules for reductive algebraic groups: characters and support varieties

TL;DR

The notes address the problem of understanding characters and support varieties of tilting modules for reductive groups over fields of positive characteristic. The authors develop and apply a -canonical framework, connecting representation-theoretic questions to the geometry of the nilpotent cone via the Springer resolution and the Lusztig–Vogan bijection, with a focus on tilting modules as a bridge to simple-character information. They prove a tilting-character formula in terms of antispherical -Kazhdan–Lusztig polynomials, and establish large-p results that align with Andersen–Jantzen–Soergel style predictions, while also pursuing more geometric approaches through the Finkelberg–Mirković conjecture and categorical structures. The work then extends to support varieties, proving Humphreys-type conjectures in significant cases, and articulates a rich program linking cohomology of Frobenius kernels, perverse-coherent sheaves, and tensor-ideal structures, with a coherent Lusztig–Vogan framework across characteristics. Together, these results illuminate the deep interplay between modular representation theory, geometric representation theory, and categorical methods, offering both concrete formulas and a broad geometric paradigm for future progress.

Abstract

These notes are our contribution to the Proceedings of the ICM 2026. We discuss some results we have obtained (in part jointly with coauthors) regarding the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. These statements mainly concern tilting modules, in particular their characters and support varieties.

Paper Structure

This paper contains 28 sections, 3 theorems, 51 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Representations of reductive algebraic groups and Kazhdan--Lusztig combinatorics of the affine Weyl group
  3. Characters
  4. Support varieties
  5. Characters
  6. Representations of reductive algebraic groups and their characters
  7. Induced, simple and tilting modules
  8. The Weyl character formula
  9. Linkage principle
  10. Translation principle
  11. Affine Hecke algebra and Kazhdan--Lusztig basis
  12. Tilting character formula
  13. Characters of simple modules: Lusztig's formula
  14. The Finkelberg--Mirković conjecture
  15. Support varieties
  16. ...and 13 more sections

Key Result

Theorem 2.6

Let $J \subset S_{\mathrm{aff}}$ be a finitary subset such that $C_J \neq \varnothing$. For any $y,w \in W_{\mathrm{aff}}^J$ we have

Theorems & Definitions (22)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Conjecture 2.8
  • Remark 2.9
  • Conjecture 2.10
  • ...and 12 more