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On the derived Lusztig correspondence

Gérard Laumon, Emmanuel Letellier

Abstract

Let $G$ be a connected reductive group, $T$ a maximal torus of $G$, and $N$ the normalizer of $T$ in $G$. In this paper we study the connection between the derived category of l-adic sheaves on the stack $[Lie(T)/N]$ and the derived category of $l$-adic sheaves on $[Lie(G)/G]$.

On the derived Lusztig correspondence

Abstract

Let be a connected reductive group, a maximal torus of , and the normalizer of in . In this paper we study the connection between the derived category of l-adic sheaves on the stack and the derived category of -adic sheaves on .

Paper Structure

This paper contains 32 sections, 48 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Postnikov diagrams in triangulated categories
  4. Stacks and sheaves : notation and convention
  5. Cohomology of $\mathrm{B}(T)$
  6. $W$-equivariant complexes
  7. Cohomological correspondences
  8. Reductive groups
  9. Lusztig correspondence
  10. Lusztig induction and restriction
  11. Induction and restriction for perverse sheaves
  12. Springer action
  13. Steinberg stacks
  14. Geometry of Steinberg stacks
  15. Purity of cohomology
  16. ...and 17 more sections

Key Result

Theorem 1.0.1

For $\lambda\in\frak{L}$, the restriction ${\mathcal{N}}_\lambda$ of ${\mathcal{N}}$ to $(\mathcal{T}\times_\mathfrak{car}\mathcal{G})_\lambda$ descends to ${\overline{{\mathcal{N}}}}_\lambda$ on $({\overline{\mathcal{T}}}\times_\mathfrak{car}\mathcal{G})_\lambda$.

Theorems & Definitions (96)

  • Theorem 1.0.1: Descent
  • Remark 1.0.2
  • Theorem 1.0.3
  • Remark 1.0.4
  • Theorem 1.0.5
  • Lemma 2.1.1
  • proof
  • Remark 2.1.2
  • Lemma 2.1.3
  • proof
  • ...and 86 more