Hypoelliptic Laplacian and twisted trace formula

Liu, Bingxiao

doi:10.1016/j.crma.2018.11.010

Cited by 5 publications

(4 citation statements)

References 44 publications

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“…Proof. By [3, Theorem 7.9.1], [28, Theorem 5.4], and use instead (4.1.20), we get that under the assumptions in this theorem, for t > 0, (4.1.22)Tr s [(N Λ • (T * Z) − m 2 ) exp(−tD Z,F,2 ))] = 0.The proof of (4.1.21) is identical to the proof of[25, Proposition 6.5.3].…”

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confidence: 82%

“…A generalization of Bismut's formula (3.3.25) to the twisted case is obtained in [24,25]. An extension of this formula for considering arbitrary elements in the center of enveloping algebra instead of Casimir operator (3.2.7) was obtained in [7] by Bismut and Shen.…”

Section: 1mentioning

confidence: 99%

“…Bergeron and Venkatesh [2] also studied the asymptotic analytic torsion but under a totally different setting. In [25,Section 7], the asymptotic equivariant analytic torsion for a locally symmetric space was studied, and the oscillating terms also appeared naturally in that case.…”

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confidence: 99%

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On full asymptotics of analytic torsions for compact locally symmetric orbifolds

Liu¹

2020

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In this paper, we consider a certain sequence of homogeneous flat vector bundles on a compact locally symmetric orbifold, and we compute explicitly the associated asymptotic Ray-Singer analytic torsion. The basic idea is converting this question via the Selberg's trace formula into computing the semisimple orbital integrals. Then the central part is to evaluate the elliptic orbital integrals which are not identity orbital integrals. 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 a smaller Lie group. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations.By (3.5.8), (3.5.9), we get (3.5.11)Recall that the measures dz, dv on Z(γ), Z(γ)\G are defined in (3.3.8). Then(3.5.12)By (3.3.7), (3.3.9), we get (3.5.13)Take γ ∈ Γ. Let K(γ) be a maximal compact subgroup of Z(γ) so that X(γ) = Z(γ)/K(γ). Then K(γ) acts on Z(γ) on the right, which induces an action on Γ ∩ Z(γ)\Z(γ) on the right. Set (3.5.14) ∆(γ) = ker(K(γ) → Diffeo(Γ ∩ Z(γ)\Z(γ))). Then ∆(γ) is a finite subgroup of Γ ∩ K(γ). Set (3.5.15) S(γ) = ker(Γ ∩ Z(γ) → Diffeo(X(γ))). Then ∆(γ) ⊂ S(γ) and S(γ) represents the isotropy group of the principal orbit type for the right action of K(γ) on Γ ∩ Z(γ)\Z(γ). As in (3.4.14), we have (3.5.16) Vol(Γ ∩ Z(γ)\Z(γ)) = Vol(K(γ)) |S(γ)| Vol(Γ ∩ Z(γ)\X(γ)).

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confidence: 82%

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confidence: 99%

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On full asymptotics of analytic torsions for compact locally symmetric orbifolds

Liu¹

2020

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“…This paper is mainly the first part of the author's thesis [34], and the main results were announced in [35].…”

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confidence: 99%

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

Liu

2021

Journal of Functional Analysis

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Hypoelliptic Laplacian and twisted trace formula

Cited by 5 publications

References 44 publications

On full asymptotics of analytic torsions for compact locally symmetric orbifolds

On full asymptotics of analytic torsions for compact locally symmetric orbifolds

Hypoelliptic Laplacian and twisted trace formula

Asymptotic equivariant real analytic torsions for compact locally symmetric spaces

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