Semi-Orthogonal Decompositions for Rank Two Imprimitive Reflection Groups

Abstract

For every imprimitive complex reflection group of rank 2, we construct a semi-orthogonal decomposition of the derived category of the associated global quotient stack which categorifies the usual decomposition of the orbifold cohomology indexed by conjugacy classes. This confirms a conjecture of Polishchuk and Van den Bergh in these cases. This conjecture was recently also proved by Ishii and Nimura for arbitrary complex reflection groups of rank 2 and real reflection groups of rank 3, but our approach is very different.