Elliptic and transversally elliptic index theory from the viewpoint of $KK$-theory

Abstract

The main purpose of the paper is to develop a $KK$-theory approach to index theory of operators transversally elliptic with respect to a proper group action. The principal novelty is the use of fields of Clifford algebras along the orbits. Index formulas that we prove allow to calculate index from symbol and vice-versa using $KK$-product. This has a deeper meaning: there exists Poincaré duality between the symbol and the index when they are defined as elements of the appropriate $KK$-groups.

The main results on transversally elliptic operators are contained in the second half of the paper. In the first part we develop the necessary technique and prove index theorems for elliptic operators. This is done in the equivariant setting with respect to a proper group action, the manifolds and groups are not assumed to be compact. At the end of the paper we give applications of our methods to transversally elliptic operators on singular foliations.