Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graded Hecke algebras, constructible sheaves and the p-adic Kazhdan--Lusztig conjecture

Maarten Solleveld

TL;DR

The paper develops a geometric framework linking twisted graded Hecke algebras to equivariant constructible sheaves on nilpotent cones, via formal completion at central characters and localization on fixed-point varieties. Standard modules are constructed geometrically and realized through endomorphisms of localized complexes, enabling a Kazhdan–Lusztig-type multiplicity formula that equates irreducible-in-standard multiplicities with local-system appearances in IC-sheaves. This yields a p-adic Kazhdan–Lusztig conjecture for representations of reductive $p$-adic groups, under plausible Langlands-parameter compatibility, and confirms the conjecture in broad classes of groups. The approach emphasizes parabolic induction compatibility and the interplay between Langlands parameters and equivariant geometry, providing a pathway toward geometrization and categorification of local Langlands correspondence. Overall, the work offers concrete mechanisms to compute multiplicities in geometric terms and to transfer KL-type results to $p$-adic settings via central-character localization.

Abstract

Graded Hecke algebras can be constructed in terms of equivariant cohomology and constructible sheaves on nilpotent cones. In earlier work, their standard modules and their irreducible modules where realized with such geometric methods. We pursue this setup to study properties of module categories of (twisted) graded Hecke algebras, in particular what happens geometrically upon formal completion with respect to a central character. We prove a version of the Kazhdan--Lusztig conjecture for (twisted) graded Hecke algebras. It expresses the multiplicity of an irreducible module in a standard module as the multiplicity of an equivariant local system in an equivariant perverse sheaf. This is applied to smooth representations of reductive p-adic groups. Under some conditions, we verify the p-adic Kazhdan--Lusztig conjecture from [Vogan]. Here the equivariant constructible sheaves live on certain varieties of Langlands parameters. The involved conditions are checked for substantial classes of groups and representations.

Graded Hecke algebras, constructible sheaves and the p-adic Kazhdan--Lusztig conjecture

TL;DR

The paper develops a geometric framework linking twisted graded Hecke algebras to equivariant constructible sheaves on nilpotent cones, via formal completion at central characters and localization on fixed-point varieties. Standard modules are constructed geometrically and realized through endomorphisms of localized complexes, enabling a Kazhdan–Lusztig-type multiplicity formula that equates irreducible-in-standard multiplicities with local-system appearances in IC-sheaves. This yields a p-adic Kazhdan–Lusztig conjecture for representations of reductive -adic groups, under plausible Langlands-parameter compatibility, and confirms the conjecture in broad classes of groups. The approach emphasizes parabolic induction compatibility and the interplay between Langlands parameters and equivariant geometry, providing a pathway toward geometrization and categorification of local Langlands correspondence. Overall, the work offers concrete mechanisms to compute multiplicities in geometric terms and to transfer KL-type results to -adic settings via central-character localization.

Abstract

Graded Hecke algebras can be constructed in terms of equivariant cohomology and constructible sheaves on nilpotent cones. In earlier work, their standard modules and their irreducible modules where realized with such geometric methods. We pursue this setup to study properties of module categories of (twisted) graded Hecke algebras, in particular what happens geometrically upon formal completion with respect to a central character. We prove a version of the Kazhdan--Lusztig conjecture for (twisted) graded Hecke algebras. It expresses the multiplicity of an irreducible module in a standard module as the multiplicity of an equivariant local system in an equivariant perverse sheaf. This is applied to smooth representations of reductive p-adic groups. Under some conditions, we verify the p-adic Kazhdan--Lusztig conjecture from [Vogan]. Here the equivariant constructible sheaves live on certain varieties of Langlands parameters. The involved conditions are checked for substantial classes of groups and representations.

Paper Structure

This paper contains 11 sections, 32 theorems, 172 equations.

Table of Contents

  1. The setup from SolSGHA
  2. Formal completion at a central character
  3. Localization on $\mathfrak g$ and $\mathfrak g_N$
  4. Twisted graded Hecke algebras with a fixed $r$
  5. Standard modules of twisted graded Hecke algebras
  6. Relation with automorphisms of $\mathbb H$
  7. Construction from $K_{\sigma,r}$ or $K_{N,\sigma,r}$
  8. Structure of the localized complexes $K_{\sigma,r}$ and $K_{N,\sigma,r}$
  9. The Kazhdan--Lusztig conjecture
  10. Localization in equivariant cohomology
  11. Compatibility with parabolic induction

Key Result

Theorem A

(see Theorem thm:2.5) There is a natural algebra isomorphism This induces an equivalence of categories Here $\mathrm{Mod}_{\mathrm{fl}, \sigma ,r}$ denotes the category of finite length modules all whose irreducible subquotients admit the central character $(\sigma,r)$.

Theorems & Definitions (57)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 47 more