Title: An explicit derived McKay correspondence for some complex reflection groups of rank 2
Abstract: Let G be a finite subgroup of $SL_2(\mathbb{C})$. The classical McKay correspondence draws connections between the representation theory of G, the geometry of the minimal resolution of singularities of $\mathbb{C}^2/G$, and the algebra of the invariant ring $\mathbb{C}[x,y]^G$. Thanks to Kapranov-Vasserot, it can also be interpreted as an equivalence of triangulated categories between the bounded derived category of coherent sheaves on the minimal resolution of \mathbb{C}^2/G and the bounded derived category of G-equivaraint sheaves on $\mathbb{C}^2$. If we enlarge to consider finite subgroups of $GL_2(\mathbb{C})$, variants of these results are known when G contains no reflections. In this talk, I will discuss recent results, inspired by work of Kawamata and Potter, giving a semiorthogonal decomposition of the derived category of G-equivariant sheaves on $\mathbb{C}^2$ whose components are in bijection with the irreducible representations of G, where G belongs to a large class of complex reflection groups of rank 2. The key input is explicit computations of group actions on the G-Hilbert schemes of Ito-Nakamura. This is joint work with Anirban Bhaduri, Yael Davidov, Eleonore Faber, Katrina Honigs, Eric Overton-Walker, and Dylan Spence.