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

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 $\tilde{\mathcal{N}}$ , can be upgraded to a ring isomorphism by a twist.