The purpose of this paper is to give a new proof of the results of Moscovici and Stanton on 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 that uses 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 a suitable Itô calculus associated with certain hypoelliptic diffusions.
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Jean-Michel Bismut, Eta invariants and the hypoelliptic Laplacian. J. Eur. Math. Soc. 21 (2019), no. 8, pp. 2355–2515DOI 10.4171/JEMS/887