An index theorem for higher orbital integrals

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 $C^{\ast }$-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.

40 pages; updates based on referee comments; expanded proof of Proposition 3.3

References in corpus (2)

• Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C*-algebra
• $G$-Homotopy Invariance of the Analytic Signature of Proper Co-compact $G$-manifolds and Equivariant Novikov Conjecture

Cited by in corpus (2)

• Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
• Higher orbital integrals, rho numbers and index theory