Eta invariants and the hypoelliptic Laplacian

TL;DR

This paper introduces a novel proof of results on eta invariants on compact locally symmetric spaces by combining hypoelliptic Laplacian techniques, Clifford algebra rotations, and probabilistic methods, offering new insights into orbital integrals.

Contribution

It presents a new proof of existing results using hypoelliptic Laplacian and probabilistic methods, diverging from traditional harmonic analysis approaches.

Findings

New proof of orbital integral results for eta invariants

Integration of hypoelliptic Laplacian with Clifford algebra rotations

Development of Itô calculus for hypoelliptic diffusions

Abstract

The purpose of this paper is to give a new proof of results of Moscovici and Stanton on the orbital integrals associated with eta invariants on compact locally symmetric spaces. Moscovici and Stanton used methods of harmonic analysis on reductive groups. Here, we combine our approach to orbital integrals using the hypoelliptic Laplacian, with the introduction of a rotation on certain Clifford algebras. Probabilistic methods play an important role in establishing key estimates. In particular, we construct the proper It{\^o} calculus associated with certain hypoelliptic diffusions.