Perverse sheaves and t-structures on the thin and thick affine flag varieties

TL;DR

This paper explores the structure of Iwahori-equivariant perverse sheaves on affine flag varieties, establishing a Ringel duality and resolving a conjecture about convolution-exact perverse sheaves being tilting.

Contribution

It extends the description of thin affine flag variety perverse sheaves to the thick case and proves a conjecture relating convolution-exact sheaves to tilting objects.

Findings

Established Ringel duality between thin and thick affine flag sheaves.

Extended the non-commutative Springer resolution description to the thick case.

Proved that convolution-exact perverse sheaves are tilting in the Iwahori-Whittaker category.

Abstract

We study the categories $\mathrm{Perv}_{\mathrm{thin}}$ and $\mathrm{Perv}_{\mathrm{thick}}$ of Iwahori-equivariant perverse sheaves on the thin and thick affine flag varieties associated to a split reductive group $G$. An earlier work of the first author describes $\mathrm{Perv}_{\mathrm{thin}}$ in terms of bimodules over the so-called non-commutative Springer resolution. We partly extend this result to $\mathrm{Perv}_{\mathrm{thick}}$, providing a similar description for its anti-spherical quotient. The long intertwining functor realizes $\mathrm{Perv}_{\mathrm{thick}}$ as the Ringel dual of $\mathrm{Perv}_{\mathrm{thin}}$; we point out that it shares some exactness properties with the similar functor acting on perverse sheaves on the finite-dimensional flag variety. We use this result to resolve a conjecture of Arkhipov and the first author, proving that the image in the Iwahori-Whittaker category of any convolution-exact perverse sheaf on the affine flag variety is tilting.