Asymptotic equivariant real analytic torsions for compact locally symmetric spaces

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.

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Hypoelliptic Laplacian and twisted trace formula

Liu¹

2023

Annales De l'Institut Fourier

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We give an explicit geometric formula for the twisted orbital integrals using the method of the hypoelliptic Laplacian developed by Bismut. Combining with the twisted trace formula, we can evaluate the equivariant trace of the heat operators of the Laplacians on a compact locally symmetric space. In particular, we revisit the equivariant local index theorems and twisted L 2 -torsions for locally symmetric spaces.Résumé. -On donne une formule géométrique explicite pour les intégrales orbitales semisimples tordues du noyau de la chaleur sur un espace symétrique, en utilisant la méthode du laplacien hypoelliptique développée par Bismut. Alors en combinant avec la formule des traces tordue, on peut évaluer les traces équivariantes de l'opérateur de la chaleur du laplacien sur un espace localement symétrique compact. En particulier, on revisite les théorèmes de l'indice équivariant local et de la torsion L 2 équivariante pour les espaces localement symétriques.

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Asymptotic equivariant real analytic torsions for compact locally symmetric spaces

Cited by 3 publications

References 40 publications

Toeplitz operators and the full asymptotic torsion forms

Toeplitz operators and the full asymptotic torsion forms

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

Hypoelliptic Laplacian and twisted trace formula

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