An index theorem for higher orbital integrals

TL;DR

This paper establishes an index theorem linking higher orbital integrals and equivariant indices of elliptic operators, providing a topological formula that fully characterizes these indices in the context of reductive Lie groups.

Contribution

It introduces a new index formula for pairings of cyclic cocycles with equivariant indices, advancing the understanding of the $K$-theory of group $C^*$-algebras for reductive Lie groups.

Findings

Derived a topological index formula for higher orbital integrals

Connected cyclic cocycles with equivariant indices of elliptic operators

Provided a method to compute equivariant indices via topological expressions

Abstract

Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions.