Asymptotic equivariant real analytic torsions for compact locally symmetric spaces
type: Item
instance of: scholarly article
title: Asymptotic equivariant real analytic torsions for compact locally symmetric spaces
zbMATH Open document ID: 1475.58025
author: Bingxiao Liu
published in: Journal of Functional Analysis
publication date: 2021-07-05T00:00:00Z
review text: The paper contains part of the author's doctoral thesis, that, in condensed form, previously appeared as [the author, C. R., Math., Acad. Sci. Paris 357, No. 1, 74--83 (2019; Zbl 1501.58011)]. It handles a natural continuation of the vast and deep work of J.-M. Bismut, who has established several results and methods of paramount importance for the subject and was the thesis advisor. In [\textit{J.-M. Bismut}, Hypoelliptic Laplacian and orbital integrals. Princeton, NJ: Princeton University Press (2011; Zbl 1234.58001)], a family of hypoelliptic Laplacians \(\mathcal{L}_b^X\) is constructed, where \(X\) is a locally symmetric space and \(b > 0\), which converges in a proper sense to a Bochner-like Laplacian \(\mathcal{L}^X\) as \(b\to 0\). Using a geometric description of the orbital integrals associated with semisimple elements in \(G\), where \(G\) is a connected real reductive Lie group, it is then obtained an explicit formula in the case of semisimple orbital integrals for the heat kernel and the wave kernel of \(\mathcal{L}^X\) (see Theorems 6.1.1 and 6.3.2 in the above mentioned book [Bismut, loc. cit.]). In this paper, the author studies the asymptotics of the equivariant analytic torsions for a certain sequence of flat vector bundles over a compact locally symmetric space. For this purpose, he introduces \(\sigma\)-twisted orbital integrals, for \(\sigma\in\Aut(G)\) and develops the technically hard work of introducing the twisted version of the tools, like the ones mentioned above, created by Bismut for the untwisted case. He shows then that the Bismut method can be adapted for the twisted version. The main result (Theorem 1.0.1) and its long proof involve the whole machinery developed along the subsequent sections. Initially, after presenting the necessary fundamental technical tools, \(\sigma\)-twisted orbital integrals are introduced (Section 2.5), followed by a central twisted trace formula (Theorem 2.6.6). In the third section of the paper, it is invest…
Mathematics Subject Classification ID: 58J52
zbMATH DE Number: 7367921
zbMATH Keywords: equivariant analytic torsion
zbMATH Keywords: locally symmetric space
zbMATH Keywords: twisted orbital integral
reviewed by: Antonio Roberto da Silva
MaRDI profile type: Publication
full work available at URL: j.jfa.2021.109117
OpenAlex ID: W3166155014
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DOI: 10.1016/J.JFA.2021.109117
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description: scientific article; zbMATH DE number 7367921