An explicit derived McKay correspondence for some complex reflection groups of rank two

Abstract: In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups $G=G(2m,m,2)$, $G_{12}$, $G_{13}$, or $G_{22}$, there is a semiorthogonal decomposition of the following form, where $B_1,\ldots,B_r$ are the normalizations of the irreducible components of the branch divisor $\mathbb{C}^2\to \mathbb{C}^2/G$ and $E_1,\ldots,E_n$ are exceptional objects: $$D^G(\mathbb{C}^2)\cong \langle E_1,\ldots,E_n,D(B_1),\ldots, D(B_r), D(\mathbb{C}^2/G)\rangle.$$ We verify that the pieces of this decomposition correspond to the irreducible representations of $G$, verifying the Orbifold Semiorthogonal Decomposition Conjecture of Polishchuk and Van den Bergh. Due to work of Potter on the group $G(m,m,2)$, this conjecture is now proven for all finite groups $G\leq \mathrm{GL}(2,\mathbb{C})$ that are generated by order $2$ reflections. Each of these groups contains, as a subgroup of index $2$, a distinct finite group $H\leq \mathrm{SL}(2,\mathbb{C})$. A key part of our work is an explicit computation of the action of $G/H$ on the $H$-Hilbert scheme $\textrm{$H$-Hilb}(\mathbb{C}^2)$.