A remark on descent for Coxeter groups

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