This talk is about an extension of the derived McKay correspondence for finite subgroups of SL(2) to complex reflection groups of rank 2 generated by reflections of order 2: The classical McKay correspondence connects the representation theory of a finite group in SL(2) and the geometry of the exceptional divisor of the minimal resolution of the corresponding quotient singularity. Kapranov and Vasserot showed that this may be realized as a derived equivalence between the derived category of coherent sheaves on the minimal resolution, and the derived category of equivariant coherent sheaves on the two-dimensional vector space the group is acting on.

On the other hand, for a complex reflection group of rank 2 generated by order 2 reflections, the quotient is smooth. Using that each such reflection group contains a distinct subgroup of SL(2) as a subgroup of index 2, we derive a semi-orthogonal decomposition of the category of equivariant coherent sheaves, where the derived category of the quotient is one piece, and the other pieces are coming from branch divisors and exceptional objects. In particular, the total number of pieces of this decomposition is equal to the number of irreducible representations of the reflection group and it can be related to the (conjectured) motivic semi-orthogonal decomposition of the derived categories of equivariant coherent sheaves for the reflection groups of Polishchuk and Van den Bergh. This is joint work with A. Bhaduri, Y. Davidov, K. Honigs, P. MacDonald, E. Overton-Walker, and D. Spence.