The index of operators on foliated bundles

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebraic Topology, but in the framework of cyclic cohomology and noncommutative geometry.Comment: AMS-TeX, 18 page

Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras

We give a detailed calculation of the Hochschild and cyclic homology of the algebra \CIc(G) of locally constant, compactly supported functions on a reductive p-adic group G. We use these calculations to extend to arbitrary elements the definition the higher orbital integrals introduced in \cite{Blanc-Brylinski} for regular semisimple elements. Then we extend to higher orbital integrals some results of Shalika. We also investigate the effect of the ``induction morphism'' on Hochschild homology.Comment: AMS-Latex, 27 page

An index theorem for families 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 \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.Comment: AMS-Latex, 39 pages, references, corrections, and new results adde

The Thom isomorphism in gauge-equivariant K-theory

In a previous paper we have introduced the gauge-equivariant K-theory group of a bundle endowed with a continuous action of a bundle of compact Lie groups. These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (i.e. a family of elliptic operators invariant with respect to the action of a bundle of compact groups). In this paper, we continue our study of gauge-equivariant K-theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge-equivariant K-theory. Then we construct push-forward maps and define the topological index of a gauge-invariant family.Comment: 29 pages, LaTeX2e, amsart, x