Rook placements and orbit harmonics

For fixed positive integers $n,m$, let ${\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times m}(\mathbb{C})$ be the affine space consisting of all $n\times m$ complex matrices, and let $\mathbb{C}[{\mathbf{x}}_{n\times m}]$ be its coordinate ring. For $0\le r\le \min \{m,n\}$, we apply the orbit harmonics method to the finite matrix loci ${\mathcal{Z}}_{n,m,r}$ of rook placements with exactly $r$ rooks, yielding a graded ${\mathfrak{S}}_n\times {\mathfrak{S}}_m$-module $R({\mathcal{Z}}_{n,m,r})$. We find one signed and two sign-free graded character formulae for $R({\mathcal{Z}}_{n,m,r})$. We also exhibit some applications of these formulae, such as proving a concise presentation of $R({\mathcal{Z}}_{n,m,r})$, and proving some module injections and isomorphisms. Some of our techniques are still valid for involution matrix loci.