Singularity categories and singular loci of certain quotient singularities

Let $V$ be a finite dimensional $k$-vector space, where $k$ is an algebraic closed field of characteristic zero. Let $G\subseteq \mathrm{S}\mathrm{L}(V)$ be a finite abelian group, and denote by $S$ the $G$-invariant subring of the polynomial ring $k[V]$. It is shown that the singularity category $D_{sg}(S)$ recovers the reduced singular locus of $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$.