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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 asymmetric, classical pseudodifferential operator $A_0$ in $\L^2(\R^n)$ thatcommutes with the regular representation of $G$, and assume that it is ellipticon $\bf{X}$. We show that the spectrum of the Friedrichs extension $A$ of theoperator $\mathrm{res} \circ A_0 \circ \mathrm{ext}: \CT({\bf{X}}) \to\L^2({\bf{X}})$ is discrete, and using the method of the stationary phase, wederive asymptotics for the number $N_\chi(\lambda)$ of eigenvalues of $A$ equalor less than $\lambda$ and with eigenfunctions in the $\chi$-isotypic componentof $\L^2({\bf{X}})$ as $\lambda \to \infty$, giving also an estimate for theremainder term for singular group actions. Since the considered critical set isa singular variety, we recur to partial desingularization in order to apply thestationary phase theorem.

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