We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed Riemannian manifold M carrying an effective and isometric action of a compact, connected Lie group G. Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it to the reduction of the corresponding Hamiltonian flow, proving that the reduced spectral counting function satisfies Weyl's law, together with an estimate for the remainder.

Let G ⊂ O(n) be a compact group of isometries acting on n-dimensional Euclidean space R n , and 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 X. We show that the spectrum of the Friedrichs extension A of the operator res • A 0 • ext : C ∞ c (X) → L 2 (X) is discrete, and using the method of the stationary phase, we derive asymptotics for the number N χ (λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ -isotypic component of L 2 (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.

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Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this paper, we study the equivariant cohomology of X = T * M , and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case that X is a compact symplectic manifold with a Hamiltonian G-action, similar residue formulae were derived by Jeffrey, Kirwan et al. [26,25].D̺(X) = d(̺(X)) − ι X (̺(X)), X ∈ g, ̺ ∈ (S(g * ) ⊗ Λ(X)) G ,where d and ι denote the usual exterior differentiation and contraction, the equivariant cohomology of the G-action on X is given by the quotient H * G (X) = Ker D/ Im D, which is canonically isomorphic to the topological equivariant cohomology introduced in [2] in case that G is compact, an assumption that we will make from now on. The main difference between ordinary and equivariant cohomology is that the latter has a larger coefficient ring, namely S(g * ), and that it depends on the orbit structure of the underlying G-action. Let us now assume that X admits a symplectic structure ω which is left invariant by G. By Cartan's homotopy formula,where L denotes the Lie derivative with respect to a vector field, implying that ι X ω is closed for each X ∈ g. G is said to act on X in a Hamiltonian fashion, if this form is even exact, meaning that there exists a linear function J : g → C ∞ (X) such that for each X ∈ g, the fundamental vector field X is equal to the Hamiltonian vector field of J(X), so that d(J(X)) + ι X ω = 0.An immediate consequence of this is that for any equivariantly closed form ̺ the form given by e i(J(X)−ω) ̺(X) is equivariantly closed, too. Following Souriau and Kostant, one defines the momentum map of a Hamiltonian action as the equivariant mapAssume next that 0 ∈ g * is a regular value of J, which is equivalent to the assumption that the stabilizer of each point of J −1 (0) is finite. In this case, J −1 (0) is a smooth manifold, and the corresponding Marsden-Weinstein reduced space, or symplectic quotientis an orbifold with a unique symplectic form ω red determined by the identity ι * ω = π * ω red , where π : J −1 (0) → X red and ι : J −1 (0) ֒→ X denote the canoncial projection and inclusion, respectively. Furthermore, π * induces an isomorphism between H * (X red ) and H * G (J −1 (0)). Consider now the mapand assume that X is compact and oriented. In this case, Kirwan [28] showed that K defines a surjective homomorphism, so that the cohomology of X red should be computable from the equivariant cohomology of X. This is the content of the residue formula of Jeffrey and Kirwan [26], which for any ̺ ∈ H * G (X) expresses the integral(1)following residue formula. Let ̺ ∈ H * G (T * M ) be of the form ̺(X) = α + Dβ(X), where α is a closed, basic differential form on T * M of compact support, and β is an equivariant differential form of compact support. Fix a maximal torus T ⊂ G, and denote the corresponding root system by ∆(g, t). Assume that the dimension κ of a principal G-orbit is equal to d = dim g, and denote the product of the pos...

Abstract. Let G C be a connected, linear algebraic group defined over R, acting regularly on a finite dimensional vector space V C over C with R-structure V R . Assume that V C posseses a Zariski-dense orbit, so that (G C , ̺, V C ) becomes a prehomogeneous vector space over R. We consider the left regular representation π of the group of R-rational points G R on the Banach space C 0 (V R ) of continuous functions on V R vanishing at infinity, and study the convolution operators π(f ), where f is a rapidly decreasing function on the identity component of G R . Denote the complement of the dense orbit by S C , and put S R = S C ∩ V R . It turns out that the restriction of π(f ) to V R − S R is a smooth operator. Furthermore, if G C is reductive, and S C and S R are irreducible hypersurfaces, π(f ) corresponds, on each connected component of V R − S R , to a totally characteristic pseudodifferential operator. We then investigate the restriction of the Schwartz kernel of π(f ) to the diagonal. It defines a distribution on V R − S R given by some power |p(m)| s of a relative invariant p(m) of (G C , ̺, V C ) and, as a consequence of the fundamental theorem of prehomogeneous vector spaces, its extension to V R , and the complex s-plane, satisfies functional equations. A trace of π(f ) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.