Recollement for perverse sheaves on real hyperplane arrangements

Abstract

We consider a hyperplane arrangement in $\mathbb{C}^n$ defined over $\mathbb{R}$, and the associated natural stratification of $\mathbb{C}^n$. The category of perverse sheaves smooth with respect to this stratification was described by Kapranov and Schechtman in terms of quiver representations. Using work of Weissman, we reinterpret this category as the category of finite-dimensional modules over an explicit algebra. We also describe recollement (open-closed decomposition) of perverse sheaves in terms of this module category. As an application, we identify the modules associated to all intersection cohomology complexes. We also compute recollement for $W$-equivariant perverse sheaves for the reflection arrangement of a finite Coxeter group $W$. We identify the equivariant intersection cohomology sheaves arising as intermediate extensions of local systems on the open stratum, thereby answering a question of Weissman.