Faisceaux caract\`eres sur les espaces de lacets d'alg\`ebres de Lie

TL;DR

This paper develops foundational results on the affine Grothendieck-Springer fibration, including constructibility, symmetry, perversity, and homotopy properties, with implications for affine Springer fibers and Hitchin fibrations.

Contribution

It introduces new constructibility and perversity results for the affine Grothendieck-Springer sheaf and extends homotopy techniques to affine Springer fibers.

Findings

Constructibility results for affine Grothendieck-Springer sheaf

Perversity statements for derived coinvariants

Homotopy results applicable to Hitchin fibration

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.