Recollement for perverse sheaves on real hyperplane arrangements

Abstract

Consider the category of perverse sheaves on Cn smooth with respect to the stratification arising from a real hyperplane arrangement. We construct an algebra whose module category is equivalent to this category, building on work of Kapranov–Schechtman. We prove that the standard recollement of perverse sheaves is equivalent to a well-known recollement for algebras.As an application of our results, we give a new description of the category of representations of the fundamental group of the complement of such a hyperplane arrangement. We also identify the modules associated to all simple perverse sheaves, that is, the intersection cohomology complexes. Finally, we generalise our results to W-equivariant perverse sheaves for the reflection arrangement of a finite Coxeter group W, extending work of Weissman. As an application, we identify the modules associated to the equivariant simple perverse sheaves.