An Index Theorem for Families 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. In this paper we concentrate on the issues specific to the case when G is trivial, so the action reduces to the action of a Lie group G. For G 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 families of invariant operators, if the bundles are trivial. We discuss then two applications, one to higher-eta invariants, which are morphisms Kn(Ψ∞inv(Y )) → C, and the other one to Fredholm boundary conditions on a simplex. As ann application of our formalism with traces, we obtain also 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.