Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let $G\subset \text{O}(n)$ be a compact group of isometries acting on $n$-dimensional Euclidean space ${\mathbb{R}}^n$, and $\mathbf{X}$ a bounded domain in ${\mathbb{R}}^n$ which is transformed into itself under the action of $G$. Consider a symmetric, classical pseudodifferential operator $A_0$ in ${\text{L}}^2({\mathbb{R}}^n)$ that commutes with the regular representation of $G$, and assume that it is elliptic on $\mathbf{X}$. We show that the spectrum of the Friedrichs extension $A$ of the operator $$\mathrm{r}\mathrm{e}\mathrm{s}$ \circ A_0 \circ $\mathrm{e}\mathrm{x}\mathrm{t}$: \CT({\bf{X}}) \to \L^2({\bf{X}})$ is discrete, and using the method of the stationary phase, we derive asymptotics for the number $N_{\chi }(\lambda )$ of eigenvalues of $A$ equal or less than $\lambda$ and with eigenfunctions in the $\chi$-isotypic component of ${\text{L}}^2(\mathbf{X})$ as $\lambda \to \mathrm{\infty }$, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.