An index theorem for families invariant with respect to a bundle of Lie groups

In [35] we defined the gauge-equivariant index of a family of ellip-tic operators invariant with respect to the free action of a bundle G → B of Lie groups and proved an index formula when the fibers of G → B are simply-connected solvable groups. In this paper we study traces on the corresponding algebras of invariant families of pseudodifferential operators and obtain a lo-cal index formula using the Fedosov product. For topologically non-trivial bundles we have to use methods of non-commutative geometry. We discuss then as an application the construction of “higher-eta invariants,” which are morphisms Kn(Ψ∞inv(Y)) → C. The algebras of invariant pseudodifferential operators that we study, ψ∞inv(Y) and Ψ inv(Y), are generalizations of “parameter dependent ” algebras of pseudodifferential operators (with parameter in Rq), so our results provide also assymptotics and an index theorem for elliptic, parameter dependent pseudodifferential operators