167 research outputs found
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
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