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

TL;DR

This paper develops an index theorem for families of elliptic operators invariant under a bundle of Lie groups, extending classical results using non-commutative geometry and providing formulas for the Chern character and local index.

Contribution

It introduces the equivariant family index for invariant elliptic operators and computes its Chern character via an Atiyah-Singer type formula, incorporating non-commutative geometry methods.

Findings

Computed the Chern character of the equivariant family index.

Derived a local index formula using the Fedosov product.

Extended index theory to parameter-dependent pseudodifferential operators.

Abstract

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ 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 $K_n(\PsS {\infty}Y) \to \CC$. The algebras of invariant pseudodifferential operators that we study, $\Psm {\infty}Y$ and $\PsS {\infty}Y$, are generalizations of ``parameter dependent'' algebras of pseudodifferential operators (with parameter in $\RR^q$), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators.