A remark on descent for Coxeter groups

TL;DR

This paper studies the descent of equivariant sheaves in finite Coxeter groups, establishing conditions under which descent to the quotient space can be inferred from descent to simpler subquotients.

Contribution

It proves that a $ ext{Gamma}$-equivariant sheaf on the reflection representation descends to the quotient if it descends to each simple reflection subgroup quotient.

Findings

Descent to the quotient space is determined by descent to simple reflection subgroups.

Provides a criterion for sheaf descent in Coxeter group actions.

Enhances understanding of equivariant sheaves in geometric representation theory.

Abstract

Let $\Gamma$ be a finite Coxeter group with reflection representation $R$. We show that a $\Gamma$-equivariant quasicoherent sheaf on $R$ descends to the quotient space $R//\Gamma$ if it descends to the quotient space $R//\langle s_i\rangle$ for every simple reflection $s_i\in \Gamma$.