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

Let $G\subset \O(n)$ be a compact group of isometries acting on $n$-dimensional Euclidean space $\R^n$, and ${\bf{X}}$ a bounded domain in $\R^n$ which is transformed into itself under the action of $G$. Consider a symmetric, classical pseudodifferential operator $A_0$ in $\L^2(\R^n)$ that commutes with the regular representation of $G$, and assume that it is elliptic on $\bf{X}$. We show that the spectrum of the Friedrichs extension $A$ of the operator $\mathrm{res} \circ A_0 \circ \mathrm{ext}: \CT({\bf{X}}) \to \L^2({\bf{X}})$ 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 $\L^2({\bf{X}})$ as $\lambda \to \infty$, 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.