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

Let G ⊂ O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in L2(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X.We show that the spectrum of the Friedrichs extension A of the operator res ◦ A0 ◦ ext :C∞ c (X)→L2(X) is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ (λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L2(X) as λ→∞, 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