A higher index on finite-volume locally symmetric spaces

TL;DR

This paper extends index theorems for Dirac operators on locally symmetric spaces to higher K-theory and broader cases, including torsion and higher rank groups, providing new computable invariants.

Contribution

It develops a higher index theorem for Dirac operators on G/K in K-theory, generalizing previous results to cases with torsion and higher rank groups.

Findings

Extended index theorems to higher K-theory.

Applicable to torsion cases and higher rank groups.

Provided computable invariants via higher orbital integrals.

Abstract

Let $G$ be a connected, real semisimple Lie group. Let $K<G$ be maximal compact, and let $\Gamma < G$ be discrete and such that $\Gamma \backslash G$ has finite volume. If the real rank of $G$ is $1$ and $\Gamma$ is torsion-free, then Barbasch and Moscovici obtained an index theorem for Dirac operators on the locally symmetric space $\Gamma \backslash G/K$. We obtain a higher version of this, using an index of Dirac operators on $G/K$ in the $K$-theory of an algebra on which the conjugation-invariant terms in Barbasch and Moscovici's index theorem define continuous traces. The resulting index theorems also apply when $\Gamma$ has torsion. The cases of these index theorems for traces defined by semisimple orbital integrals extend to Song and Tang's higher orbital integrals, and yield nonzero and computable results even when $\operatorname{rank}(G)> \operatorname{rank}(K)$, or the real rank of $G$ is larger than $1$.