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

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