Perverse sheaves on infinite-dimensional stacks, and affine Springer theory

TL;DR

This paper develops a new perverse t-structure for l-adic sheaves on infinite-dimensional stacks and proves the affine Grothendieck-Springer sheaf is perverse, extending classical theory to infinite-dimensional settings.

Contribution

It introduces a novel perverse t-structure on infinite-dimensional stacks and establishes the perversity of the affine Grothendieck-Springer sheaf, overcoming challenges of infinite-dimensional ind-schemes.

Findings

Constructed a perverse t-structure on the infinity-category of l-adic LG-equivariant sheaves.

Proved the affine Grothendieck-Springer sheaf is perverse.

Showed the sheaf is an intermediate extension of its restriction to regular semi-simple elements.

Abstract

The goal of this work is to construct a perverse t-structure on the infinity-category of l-adic LG-equivariant sheaves on the loop Lie algebra Lg and to show that the affine Grothendieck-Springer sheaf S is perverse. Moreover, S is an intermediate extension of its restriction to the locus of ``compact" elements with regular semi-simple reduction. Note that classical methods do not apply in our situation because LG and Lg are infinite-dimensional ind-schemes.