A Weyl's Law for Singular Riemannian Foliations with Applications to Invariant Theory

We prove a version of Weyl’s Law for the basic spectrum of a closed singular Riemannian foliation $(M,\mathcal{F})$ with basic mean curvature. In the special case of $M={\mathbb{S}}^n$, this gives an explicit formula for the volume of the leaf space ${\mathbb{S}}^n\mathrm{/}\mathcal{F}$ in terms of the algebra of basic polynomials. In particular, $\mathrm{Vol}({\mathbb{S}}^n\mathrm{/}\mathcal{F})$ is a rational multiple of $\mathrm{Vol}({\mathbb{S}}^m)$, where $m=\dim ({\mathbb{S}}^n\mathrm{/}\mathcal{F})$.