Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components

Abstract

Let Γ be a finite group acting on a smooth, compact manifold M , let P ∈ ψ m (M ; E 0 , E 1 ) be a Γ-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles E i → M , i = 0, 1, and let α be an irreducible representation of the group Γ. Then P induces 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 prove that the map πα(P ) is Fredholm if, and only if, P is α-elliptic, an explicit condition that we define in terms of the principal symbol of P and the action of Γ on the vector bundles E i . The result is not true for non-discrete groups. In the process, we also obtain several results on the structure of the algebra of invariant pseudodifferential operators on E 0 ⊕ E 1 , especially in relation to induced representations. We include applications to Hodge theory and to index theory of singular quotient spaces. M 3.2. The restriction morphisms 3.3. Local calculations 3.4. Calculations for the principal orbit bundle 3.5. The non-principal orbit case 4. Applications and extensions 4.1. Reduction to the connected case and α-ellipticity 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). V.N. was also supported by the NSF grant DMS 1839515.Manuscripts available from http://www.math.psu.edu/nistor/.