Equivariant perverse sheaves on Coxeter arrangements and buildings

When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${$\mathsf{P}\mathsf{e}\mathsf{r}\mathsf{v}$}W(E{\mathbb C}, {$\mathcal{H}$}{\mathbb C})$of$W$-equivariantperversesheaveson$E{\mathbb C}$, 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 ${\mathsf{P}\mathsf{e}\mathsf{r}\mathsf{v}}_W(E_{\mathbb{C}},{\mathcal{H}}_{\mathbb{C}})$ 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.