Lifting derived equivalences of abelian surfaces to generalized Kummer varieties

In this article, we study the $G$-autoequivalences of the derived category ${\mathbf{D}}_G^b(A)$ of $G$-equivariant objects for an abelian variety $A$ with $G$ being a finite subgroup of ${\mathrm{P}\mathrm{i}\mathrm{c}}^0(A)$. We provide a result analogue to Orlov’s short exact sequence for derived equivalences of abelian varieties. It can be generalized to the derived equivalences of abelian varieties for a same $G$ in general. Furthermore, we find derived equivalences of generalized Kummer varieties by lifting derived equivalences of abelian surfaces using the $G$-equivariant version of Orlov’s short exact sequence and some ``splitting" propositions.