An Index Theorem for Families of Elliptic Operators Invariant with Respect to a Bundle of Lie Groups

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle G of Lie groups. If the fibers of G → B are simply-connected solvable, we then compute the Chern character of the (equivariant family) index, the result being given by an Atiyah-Singer type formula. We also study traces on the corresponding algebras of pseudodifferential operators and obtain a local index formula for such families of invariant operators, 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. We also obtain new proofs of the regularity at s = 0 of η(D0, s), the eta function of D0, and of the relation η(D0, s) = πTr1(DD) (here D = D0 + ∂t, D = [D, t]). 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 R), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators.