Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

Abstract

We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let Γ be a compact group acting on a smooth, compact, manifold M without boundary and let P ∈ ψ m (M ; E 0 , E 1 ) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles E 0 and E 1 . Let α be an irreducible representation of the group Γ. Then P induces by restriction a map πα(P ) : H s (M ; E 0 )α → H s−m (M ; E 1 )α between the α-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map πα(P ) to be Fredholm. It turns out that the discrete and non-discret cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. Moreover, some results are true only in the abelian case. We thus concentrate in this paper on the case when Γ is finite abelian. We prove then that the restriction πα(P ) is Fredholm if, and only if, P is "α-elliptic," a condition defined in terms of the principal symbol of P (Definition 1.1). If P is elliptic, then P is also α-elliptic, but the converse is not true in general. However, if Γ acts freely on a dense open subset of M , then P is α-elliptic for the given fixed α if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra ψ m (M ; E) Γ of classical, Γ-invariant pseudodifferential operators acting on sections of the vector bundle E → M and of the structure of its restrictions to the isotypical components of Γ. These structures are described in terms of the isotropy groups of the action of the group Γ on E → M . M.L. was partially supported by the Hausdorff Center for Mathematics, Bonn. A.B., R.C., and V.N. have been partially supported by ANR-14-CE25-0012-01 (SINGSTAR). Manuscripts available from http://www.math.psu.edu/nistor/. 1 2 A. BALDARE, R. C ÔME, M. LESCH, AND V. NISTOR 2.5. Pseudodifferential operators 13 2.6. Reduction to order-zero operators 13 3. The structure of regularizing operators 14 3.1. Inner actions of Γ: the abstract case 14 3.2. Inner actions of Γ: pseudodifferential operators 15 3.3. The case of free actions 16 4. The principal symbol 17 4.1. The primitive ideal spectrum of the symbol algebra 17 4.2. Factoring out the minimal isotropy 19 5. Applications and extensions 21 5.1. Fredholm conditions 21 5.2. Boundary value problems 22 5.3. The case of non-discrete groups 23 References 23