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Faisceaux caractères sur les espaces de lacets d'algèbres de Lie

Alexis Bouthier

Abstract

We establish several foundational results regarding the Grothendieck-Springer affine fibration. More precisely, we prove some constructibility results on the affine Grothendieck-Springer sheaf and its coinvariants, enrich it with a group of symmetries, analog to the situation of Hitchin fibration, prove some perversity statements once we take some derived coinvariants and construct some specialization morphisms for the homology of affine Springer fibers. Along the way, we prove some homotopy result on l-adic complexes that can also be applied to Hitchin fibration.

Faisceaux caractères sur les espaces de lacets d'algèbres de Lie

Abstract

We establish several foundational results regarding the Grothendieck-Springer affine fibration. More precisely, we prove some constructibility results on the affine Grothendieck-Springer sheaf and its coinvariants, enrich it with a group of symmetries, analog to the situation of Hitchin fibration, prove some perversity statements once we take some derived coinvariants and construct some specialization morphisms for the homology of affine Springer fibers. Along the way, we prove some homotopy result on l-adic complexes that can also be applied to Hitchin fibration.

Paper Structure

This paper contains 57 sections, 12 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Préambule
  3. Rappels sur les résultats de BKV
  4. Les prolongements de BKV
  5. Symétries locales et perversité
  6. Constructibilité
  7. Lemme d'homotopie et perversité des coinvariants
  8. La situation fibre à fibre de Yun
  9. L'énoncé $\infty$-catégorique au niveau des complexes
  10. Morphismes de spécialisation
  11. Remerciements
  12. Fibration de Grothendieck-Springer affine
  13. Quotient adjoint et centralisateur régulier
  14. Théorème de Chevalley
  15. Centralisateur régulier
  16. ...and 42 more sections

Key Result

Proposition 8

Soit $J_{\mathfrak{b}}$ le tiré-en-arrière à $\mathfrak{b}=\mathop{\mathrm{Lie}}\nolimits(B)$, soit $C_{\mathfrak{b}}$ le schéma des centralisateurs au-dessus de $\mathfrak{b}$ pour l'action adjointe de $B$. Alors on a un morphisme canonique: de telle sorte que $j_{\mathfrak{g},\vert\mathfrak{b}}$ se factorise en :

Theorems & Definitions (45)

  • Conjecture 6
  • Proposition 8
  • proof
  • proof
  • proof
  • Proposition 29
  • proof
  • Proposition 30
  • proof
  • proof
  • ...and 35 more