Hypoelliptic Laplacian and twisted trace formula
Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 1909-1985.

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.

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.

Received:
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Accepted:
Published online:
DOI: 10.5802/aif.3566
Classification: 22E30, 53C35, 58J20, 58J50
Keywords: Twisted orbital integral, Casimir operator, Hypoelliptic Laplacian, Symmetric space
Mot clés : Intégrale orbitale tordue, Opérateur de Casimir, Laplacien hypoelliptique, Espace symétrique
Liu, Bingxiao 1

1 Universität zu Köln Weyertal 86-90 D-50931 Köln (Deutschland)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Liu, Bingxiao. Hypoelliptic Laplacian and twisted trace formula. Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 1909-1985. doi : 10.5802/aif.3566. https://aif.centre-mersenne.org/articles/10.5802/aif.3566/

[1] Arthur, J.; Clozel, L. Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, 120, Princeton University Press, Princeton, NJ, 1989, xiv+230 pages | MR | Zbl

[2] Atiyah, M. F.; Bott, R. A Lefschetz fixed point formula for elliptic complexes. I, Ann. Math. (2), Volume 86 (1967), pp. 374-407 | DOI | MR | Zbl

[3] Atiyah, M. F.; Bott, R. A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. Math. (2), Volume 88 (1968), pp. 451-491 | DOI | MR | Zbl

[4] Bergeron, Nicolas; Lipnowski, Michael Twisted limit formula for torsion and cyclic base change, J. Éc. polytech. Math., Volume 4 (2017), pp. 435-471 | DOI | Numdam | MR | Zbl

[5] Bergeron, Nicolas; Venkatesh, A. The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu, Volume 12 (2013) no. 2, pp. 391-447 | DOI | MR | Zbl

[6] Berline, N.; Getzler, E.; Vergne, M. Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004, x+363 pages (Corrected reprint of the 1992 original) | MR | Zbl

[7] Bismut, Jean-Michel The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 379-476 | DOI | MR | Zbl

[8] Bismut, Jean-Michel The hypoelliptic Laplacian on a compact Lie group, J. Funct. Anal., Volume 255 (2008) no. 9, pp. 2190-2232 | DOI | MR | Zbl

[9] Bismut, Jean-Michel Hypoelliptic Laplacian and orbital integrals, Annals of Mathematics Studies, 177, Princeton University Press, Princeton, NJ, 2011, xii+330 pages | DOI | MR | Zbl

[10] Bismut, Jean-Michel; Lebeau, Gilles The hypoelliptic Laplacian and Ray–Singer metrics, Annals of Mathematics Studies, 167, Princeton University Press, Princeton, NJ, 2008, x+367 pages | DOI | MR | Zbl

[11] Bismut, Jean-Michel; Ma, Xiaonan; Zhang, Weiping Opérateurs de Toeplitz et torsion analytique asymptotique, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 17-18, pp. 977-981 | DOI | Numdam | MR | Zbl

[12] Bismut, Jean-Michel; Ma, Xiaonan; Zhang, Weiping Asymptotic torsion and Toeplitz operators, J. Inst. Math. Jussieu, Volume 16 (2017) no. 2, pp. 223-349 | DOI | MR | Zbl

[13] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991, xii+288 pages | DOI | MR | Zbl

[14] Borel, Armand; Wallach, Nolan R. Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical Surveys and Monographs, 67, American Mathematical Society, Providence, RI, 2000, xviii+260 pages | DOI | MR | Zbl

[15] Bouaziz, Abderrazak Formule d’inversion d’intégrales orbitales tordues, Compos. Math., Volume 81 (1992) no. 3, pp. 261-290 | Numdam | MR | Zbl

[16] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005, xii+434 pages | MR | Zbl

[17] Clozel, Laurent Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982) no. 1, pp. 45-115 | DOI | Numdam | MR | Zbl

[18] Delorme, P. Théorème de Paley–Wiener invariant tordu pour le changement de base /, Compositio Mathematica, Volume 80 (1991) no. 2, pp. 197-228 | MR | Zbl

[19] Duflo, M. Généralités sur les représentations induites, Représentations des Groupes de Lie Résolubles (Monographies de la Société Mathématique de France), Volume 4, Dunod, 1972, pp. 93-119

[20] Duistermaat, J. J.; Kolk, J. A. C. Lie groups, Universitext, Springer-Verlag, Berlin, 2000, viii+344 pages | DOI | MR | Zbl

[21] Eberlein, Patrick B. Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996, vii+449 pages | MR | Zbl

[22] Helgason, Sigurđur Differential geometry and symmetric spaces, Pure and Applied Mathematics, XII, Academic Press, New York-London, 1962, xiv+486 pages | MR | Zbl

[23] Hochschild, G. The automorphism group of Lie group, Trans. Amer. Math. Soc., Volume 72 (1952), pp. 209-216 | DOI | MR | Zbl

[24] Hörmander, Lars Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967), pp. 147-171 | DOI | MR | Zbl

[25] Hörmander, Lars The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis, Classics in Mathematics, Springer-Verlag, Berlin, 2003, x+440 pages Reprint of the second (1990) edition [Springer, Berlin] | DOI | MR | Zbl

[26] Hörmander, Lars The analysis of linear partial differential operators. III: Pseudo-differential operators, Classics in Mathematics, Springer, Berlin, 2007, viii+525 pages | DOI | MR | Zbl

[27] Humphreys, James E. Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer-Verlag, New York-Heidelberg, 1975, xiv+247 pages | DOI | MR | Zbl

[28] Knapp, Anthony W. Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, 36, Princeton University Press, Princeton, NJ, 1986, xviii+774 pages | DOI | MR | Zbl

[29] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002, xviii+812 pages | MR | Zbl

[30] Kolmogoroff, A. Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. Math. (2), Volume 35 (1934) no. 1, pp. 116-117 | DOI | MR | Zbl

[31] Kostant, Bertram Clifford algebra analogue of the Hopf–Koszul–Samelson theorem, the ρ-decomposition C(𝔤)=EndV ρ C(P), and the 𝔤-module structure of 𝔤, Adv. Math., Volume 125 (1997) no. 2, pp. 275-350 | DOI | MR | Zbl

[32] Langlands, Robert P. Base change for GL(2), Annals of Mathematics Studies, 96, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980, vii+237 pages | MR | Zbl

[33] Lawson Jr., H. Blaine; Michelsohn, Marie-Louise Spin geometry, Princeton Mathematical Series, 38, Princeton University Press, Princeton, NJ, 1989, xii+427 pages | MR | Zbl

[34] Liu, Bingxiao Hypoelliptic Laplacian and twisted trace formula, Ph. D. Thesis, Université Paris-Saclay (2018) (https://tel.archives-ouvertes.fr/tel-01841334/file/72052_LIU_2018_archivage.pdf)

[35] Liu, Bingxiao Hypoelliptic Laplacian and twisted trace formula, C. R. Math. Acad. Sci. Paris, Volume 357 (2019) no. 1, pp. 74-83 | DOI | Numdam | MR | Zbl

[36] Liu, Bingxiao Asymptotic equivariant real analytic torsions for compact locally symmetric spaces, J. Funct. Anal., Volume 281 (2021) no. 7, 109117, 54 pages | DOI | MR | Zbl

[37] Lott, J. Heat kernels on covering spaces and topological invariants, J. Differential Geom., Volume 35 (1992) no. 2, pp. 471-510 | DOI | MR | Zbl

[38] Ma, X. Geometric hypoelliptic Laplacian and orbital integrals [after Bismut, Lebeau and Shen], Séminaire Bourbaki. Vol. 2016/2017. Exposés 1120–1135 (Astérisque), Volume 407, Société Mathématique de France (SMF), 2019, pp. 333-389 (Exp. No. 1130) | DOI | MR | Zbl

[39] Mathai, Varghese L 2 -analytic torsion, J. Funct. Anal., Volume 107 (1992) no. 2, pp. 369-386 | DOI | MR | Zbl

[40] McKean Jr., H. P.; Singer, I. M. Curvature and the eigenvalues of the Laplacian, J. Differ. Geom., Volume 1 (1967) no. 1, pp. 43-69 | DOI | MR | Zbl

[41] Moscovici, Henri; Stanton, Robert J. R-torsion and zeta functions for locally symmetric manifolds, Invent. Math., Volume 105 (1991) no. 1, pp. 185-216 | DOI | MR | Zbl

[42] Müller, W.; Pfaff, J. Analytic torsion and L 2 -torsion of compact locally symmetric manifolds, J. Differ. Geom., Volume 95 (2013) no. 1, pp. 71-119 | DOI | MR | Zbl

[43] Segal, Graeme The representation ring of a compact Lie group, Inst. Hautes Études Sci. Publ. Math. (1968) no. 34, pp. 113-128 | DOI | Numdam | MR | Zbl

[44] Selberg, A. On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, pp. 147-164 | MR | Zbl

[45] Shen, Shu Analytic torsion, dynamical zeta functions and orbital integrals, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 4, pp. 433-436 | DOI | Numdam | MR | Zbl

[46] Shen, Shu Analytic torsion, dynamical zeta functions, and the Fried conjecture, Anal. PDE, Volume 11 (2018) no. 1, pp. 1-74 | DOI | MR | Zbl

[47] Taylor, M. E. Pseudodifferential operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981, xi+452 pages | DOI | MR | Zbl

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