Asymptotic equivariant real analytic torsions for compact locally symmetric spaces
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 investigated the equivariant Ray-Singer analytic torsion, twisted case, compared with [\textit{J.-M. Bismut} and \textit{G. Lebeau}, The hypoelliptic Laplacian and Ray-Singer metrics. Princeton, NJ: Princeton University Press (2008; Zbl 1156.58001)] leading to a vanishing theorem on the equivariant analytic torsions (Theorem 3.3.2). Finally, the author studies the asymptotics of the equivariant real analytic torsion and shows that its leading term can be calculated in terms of the W-invariants with oscillating coefficients, as constructed for the untwisted case by Bismut-Ma-Zhang, see also [\textit{J.-M. Bismut} et al., J. Inst. Math. Jussieu 16, No. 2, 223--349 (2017; Zbl 1381.58015)]. Theorems 4.4.1 and 4.6.2 provide the estimates needed to conclude the proof of the main result of this excellent paper.
scientific article; zbMATH DE number 7367921
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