1999/06/28 by Victor Nistor, Nistor, Victor
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.math/9906188
AMS-Latex, 39 pages, references, corrections, and new results added
arxiv created 2000/08/16 · arxiv updated 2009/11/30
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 → 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(\PsS ∞Y) → \CC. The algebras of invariant pseudodifferential operators that we study, \Psm ∞Y and \PsS ∞Y, are generalizations of ``parameter dependent'' algebras of pseudodifferential operators (with parameter in \RRq), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators.