A higher index on finite-volume locally symmetric spaces
Hao Guo, Peter Hochs, Hang Wang
TL;DR
This work develops a higher index theory for Dirac operators on finite-volume locally symmetric spaces $X = \Gamma \backslash G/K$ by constructing a $K$-theoretic index in the algebra $\mathcal{A}(G)$ of exponentially decaying kernels. It extends Barbasch–Moscovici’s rank-one index to higher orbital integrals (Song–Tang) and to torsion in $\Gamma$, showing that semisimple traces may vanish in higher-rank settings while the higher index remains detectable via higher cocycles. The framework relates to the classical $\Gamma$-index through Caselman-type Schwartz algebras and a network of geometric and spectral traces, and provides an explicit holomorphic fixed-point/orbital-integral machinery to compute pairings with the index. Collectively, the results broaden index theory for proper cocompact group actions, enabling nontrivial invariants in cases where traditional semisimple traces fail to capture the index.
Abstract
Let $G$ be a connected, real semisimple Lie group. Let $K<G$ be maximal compact, and let $Γ< G$ be discrete and such that $Γ\backslash G$ has finite volume. If the real rank of $G$ is $1$ and $Γ$ is torsion-free, then Barbasch and Moscovici obtained an index theorem for Dirac operators on the locally symmetric space $Γ\backslash G/K$. 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 $Γ$ 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 $1$.