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.