Twisted equivariant HKR theorem for torus action and the small quantum group

TL;DR

This paper proves an equivariant version of the twisted Hochschild-Kostant-Rosenberg (HKR) theorem for torus actions, enabling a ring isomorphism between Hochschild cohomology of small quantum groups and sheaf cohomology on Springer resolutions.

Contribution

It extends the twisted HKR isomorphism to equivariant settings and upgrades the Bezrukavnikov-Lachowska isomorphism to a ring isomorphism with a twist.

Findings

Established equivariant twisted HKR isomorphism for torus actions.

Upgraded the Bezrukavnikov-Lachowska isomorphism to a ring isomorphism.

Connected Hochschild cohomology of small quantum groups to Springer resolution cohomology.

Abstract

We show that when a torus $T$ acts on a smooth variety $X$, the twisted HKR isomorphism is equivariant. The main consequence is that the Bezrukavnikov- Lachowska isomorphism, relating the Hochschild cohomology of the principal block of the small quantum group to certain sheaf cohomology groups on the Springer resolution $\widetilde{\mathcal{N}}$ , can be upgraded to a ring isomorphism by a twist.