K-theory and index theory on manifolds with a proper Lie group action

TL;DR

This paper develops a comprehensive index theory for orbital and transverse elliptic operators on manifolds with proper Lie group actions, correcting previous errors and introducing new results with simplified pseudo-differential operator methods.

Contribution

It provides a unified treatment of orbital and transverse index theories, correcting earlier mistakes and extending the theory with new proofs and simplified constructions.

Findings

Established a final index theorem for orbital operators.

Provided KK-theoretic proofs for various elliptic operators.

Introduced a simpler method for constructing pseudo-differential operators.

Abstract

The paper is devoted to the index theory of orbital and transverse elliptic operators on manifolds with a proper Lie group action. It corrects errors of my previous paper (published in JNCG in 2016) on transverse operators and contains new results. The two index theories, orbital and transverse, are very much intertwined and interdependent, and are treated together. The theory of orbital operators is developed from the basic definitions to the final index theorem. The KK-theoretic proofs of index theorems for elliptic, t-elliptic and orbital elliptic operators are given in sections 9, 10, 11. Throughout the paper, we use a simpler method in constructing pseudo-differential operators.