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 * -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.

An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the $K$-theoretic index

“…It has been applied to various problems in geometry and topology, such as questions about positive scalar curvature and the Novikov conjecture. In this context, the number (1.5) was shown to be relevant to representation theory, orbifold geometry and trace formulas [20,22,23,36].…”

Primary and secondary invariants of Dirac operators on $G$-proper manifolds

“…Delocalized higher indices. To improve Theorem 4.11 to allow G to be nonequal rank, in [24], Hochs, Song, and Tang computed the pairing between the cocycle Φ P g and the index element Ind C * r (G) (D). Let X/AN be the quotient of X with respect to the AN < G action.…”

“…Hochs, Song, and Tang [24] proved a reduction theorem relating the index pairing for D and the index pairing for D X/AN . 4.12.…”

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

“…There are various index theorems for the the image of the right hand side of (3.7) under traces [24,41,40] or pairings with higher cyclic cocycles [23,33,34]. Via these results, Theorem 3.4 yields topological expressions for the corresponding images of the left hand side of (3.7).…”