On full asymptotics of real analytic torsions for compact locally symmetric orbifolds

Liu, Bingxiao

doi:10.2140/apde.2024.17.1261

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On full asymptotics of real analytic torsions for compact locally symmetric orbifolds

Bingxiao Liu

Vol. 17 (2024), No. 4, 1261–1329

DOI: 10.2140/apde.2024.17.1261

Abstract

We consider a certain sequence of flat vector bundles on a compact locally symmetric orbifold, and we evaluate explicitly the associated asymptotic Ray–Singer real analytic torsion. The basic idea is to computing the heat trace via Selberg’s trace formula, so that a key point in this paper is to evaluate the orbital integrals associated with nontrivial elliptic elements. For that purpose, we deduce a geometric localization formula, so that we can rewrite an elliptic orbital integral as a sum of certain identity orbital integrals associated with the centralizer of that elliptic element. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations.

Keywords

real analytic torsion, orbital integral, orbifold, locally symmetric space

Mathematical Subject Classification

Primary: 11F72, 53C35, 57R18, 58C40

Milestones

Received: 25 July 2021

Revised: 16 August 2022

Accepted: 23 September 2022

Published: 17 May 2024

Authors

Bingxiao Liu
Mathematisches Institut Universität zu Köln Köln Germany

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Vol. 17, No. 4, 2024