# Equivariant perverse sheaves on Coxeter arrangements and buildings

## TL;DR

This paper classifies $W$-equivariant perverse sheaves on Coxeter arrangements and extends the framework to equivariant perverse sheaves on affine buildings, linking geometric representation theory with algebraic structures.

## Contribution

It provides an explicit algebraic description of equivariant perverse sheaves on Coxeter arrangements and introduces a new perspective on perverse sheaves on affine buildings.

## Key findings

- Category equivalence with modules over an explicit algebra
- Construction of equivariant perverse sheaves on affine buildings
- Connection to depth zero representations of p-adic groups

## 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.