Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case

Let $G\subset \O(n)$ be a 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 $\L^2(\R^n)$ with G-invariant Weyl symbol, and assume that it is semi-bounded from below. 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 derive asymptotics for the number $N_\chi(\lambda)$ of eigenvalues of A less or equal $\lambda$ and with eigenfunctions in the $\chi$-isotypic component of $\L^2({\bf{X}})$, giving also an estimate for the remainder term in both cases where G is a finite, or, more generally, a compact group. In particular, we show that the multiplicity of each unitary irreducible representation in $\L^2({\bf{X}})$ is asymptotically proportional to its dimension.