Equivariant perverse sheaves on Coxeter arrangements and buildings

Abstract

When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${\mathsf{Perv}}_W(E_{\mathbb{C}},{\mathcal{H}}_{\mathbb{C}})$ of $W$-equivariant perverse sheaves on $E_{\mathbb{C}}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtman's recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${\mathsf{Perv}}_W(E_{\mathbb{C}},{\mathcal{H}}_{\mathbb{C}})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations.

28 pages, 6 figures. v5 processed for publication in Epiga

References in corpus (1)

• Perverse Schobers

Cited by in corpus (1)

• Recollement for perverse sheaves on real hyperplane arrangements