The equivariant analytic index for proper groupoid actions

The paper constructs the analytic index for an elliptic pseudodifferential family of $L^{m}_{\rho,\de}$-operators invariant under the proper action of a continuous family groupoid on a $G$-compact, $C^{\infty,0}$ $G$-space.

💡 Research Summary

The paper develops a full-fledged equivariant analytic index theory for families of elliptic pseudodifferential operators that are invariant under proper actions of continuous‑family groupoids. The setting is a continuous family groupoid (G\rightrightarrows M) acting properly on a (G)-compact (C^{\infty,0}) (G)-space (X). Properness guarantees that the orbit space (X/G) is compact and that the associated reduced groupoid (C^{})-algebra (C^{}(G)) is a well‑behaved, non‑degenerate (C^{*})-algebra.

The main analytic object is a family (P={P_{x}}{x\in X}) of pseudodifferential operators belonging to the Hörmander class (L^{m}{\rho,\delta}) with order (m) and type ((\rho,\delta)) satisfying (0\le\delta<\rho\le1). Each (P_{x}) acts on the fibre (X_{x}) and the whole family is required to be (G)-invariant in the sense that for every arrow (g\in G) one has (g\cdot P_{s(g)} = P_{t(g)}\cdot g). This equivariance forces the principal symbol (\sigma(P)) to be a (G)-invariant section of the appropriate symbol bundle and to satisfy a global ellipticity condition uniformly over the groupoid.

The authors first prove that the invariant symbol is non‑degenerate on each cotangent fibre, thereby establishing global ellipticity. They then embed the analysis into the framework of Hilbert (C^{*}(G))-modules. By completing the space of smooth compactly supported sections of a vector bundle over (X) with respect to Sobolev norms, one obtains (G)-equivariant Sobolev modules (H^{s}(X)) and (H^{s-m}(X)). The operator family (P) extends to a bounded adjointable operator \