Spherical Twists for Gorenstein Orders and $G$-Hilb

Abstract

This paper constructs derived autoequivalences of Gorenstein orders as twists around spherical functors. More precisely, given a Gorenstein order $A$ and a quotient $p \colon A \to B$, then we specify natural conditions on $B$ under which the twist around the corresponding derived restriction of scalars functor is a derived autoequivalence of $A$. In the process, we show that the associated cotwist is a shift of the Nakayama functor of $B$. These results, together with local-to-global technology, are then used construct new derived autoequivalences for skew group algebras and $G$-Hilbert schemes, and we apply this theory to explicit examples.