Support singulier et homologie des fibres de Springer affines

TL;DR

This paper develops a theory of singular support for infinite dimensional stacks, applies it to affine Springer fibers, and addresses a conjecture on their homology related to root valuation stratification.

Contribution

It introduces a new theory of singular support for infinite dimensional stacks and applies it to affine Springer fibers, solving a conjecture on their homology.

Findings

Computed the singular support of the affine Springer sheaf

Established functoriality properties of the singular support theory

Confirmed the Goresky-Kottwitz-McPherson homology conjecture

Abstract

We develop a theory of singular support for various infinite dimensional stacks and establish several functoriality properties. Then we apply this theory to compute the singular support of the Grothendieck-Springer affine Springer sheaf and rely it to the local constancy conjecture of Goresky-Kottwitz-McPherson on the homology of affine Springer fibers along the root valuation stratification.