Singular equivariant asymptotics and Weyl’s law. On the distribution of eigenvalues of an invariant elliptic operator

Abstract

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.

The equivariant spectral function of an invariant elliptic operator. L -bounds, caustics, and concentration of eigenfunctions

“…could in principle be deduced from work of Donnelly [7] and Brüning-Heintze [3], at least when G acts on M with orbits of the same dimension κ. But they would not be sufficient to imply our results, and the desingularization techniques developed in [22] are necessary in order to describe the precise nature of the reduced spectral function of an invariant elliptic operator. L p -bounds for spectral clusters for elliptic second-order differential operators on 2-dimensional compact manifolds with boundary and either Dirichlet or Neumann conditions were shown in [28], while manifolds with maximal eigenfunction growth were studied in [31].…”

“…Let us assume in the following that G is a continuous group. Asymptotics for the integrals (3.3) were given in [22] using the stationary phase principle, and we will rely on these results in parts to perform a similar analysis for the integrals I x,y (µ). Write κ(x) = (x 1 , .…”

“…Let M be a closed connected Riemannian manifold and G a continuous compact Lie group acting on M by isometries. In what follows, we shall recall the construction of the partial desingularization (6.2) of the critical set C := (x, η, g) ∈ (Ω ∩ T * M ) × G | g ∈ G (x,η) performed in [22]. The desingularization process presented here is exactly the same, only that we apply it now to the study of the integrals (3.1) instead of the integrals (3.3).…”

“…Thus, the combinatorial complexity of the underlying group action is reflected in the asymptotic shape of the equivariant spectral function. By integrating the asymptotic formulae (1.8) and (1.11) This was the main result of [22]. Notice that in spite of the fact that the desingularization techniques developed there are necessary to establish the remainder estimate in (1.12), singular and exceptional orbits, being of measure zero, do not contribute to the equivariant Weyl law (1.12), and remain hidden.…”

“…If G acts on M with orbits of the same dimension, we obtain hybrid L p -bounds for eigenfunctions in the eigenvalue and isotypic aspect that improve on the classical estimates for generic eigenfunctions proved by Seeger and Sogge [29,26], but cannot hold when singular orbits are present. In the latter case, we are able to describe the caustic behaviour of the reduced spectral function as one approaches orbits of singular type, relying on our recent work [22] on singular equivariant asymptotics obtained via desingularization techniques. As an application, we are able to prove point-wise bounds for isotypic clusters of eigenfunctions, showing that they tend to concentrate on singular orbits.…”