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Geometric orbital integrals and the center of the enveloping algebra

Published online by Cambridge University Press:  11 August 2022

Jean-Michel Bismut
Affiliation:
Institut de Mathématique d'Orsay, Université Paris-Saclay, Bâtiment 307, 91405 Orsay, France jean-michel.bismut@universite-paris-saclay.fr
Shu Shen
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Case Courrier 247, 4 place Jussieu, 75252 Paris Cedex 05, France shu.shen@imj-prg.fr

Abstract

The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.

Type
Research Article
Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The authors are much indebted to Laurent Clozel for his stimulating remarks during the preparation of the paper, and for reading the preliminary version very carefully. We thank the referee for his helpful remarks.

References

Atiyah, M. F. and Bott, R., A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374407.CrossRefGoogle Scholar
Atiyah, M. F. and Bott, R., A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451491.CrossRefGoogle Scholar
Atiyah, M. F. and Schmid, W., A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 162.CrossRefGoogle Scholar
Atiyah, M. F. and Singer, I. M., The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484530.CrossRefGoogle Scholar
Atiyah, M. F. and Singer, I. M., The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546604.CrossRefGoogle Scholar
Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of nonpositive curvature, Springer Monographs in Mathematics, vol. 61 (Birkhäuser, Boston, MA, 1985).CrossRefGoogle Scholar
Berline, N., Getzler, E. and Vergne, M., Heat kernels and Dirac operators, Grundlehren Text Editions (Springer, Berlin, 2004). Corrected reprint of the 1992 original.Google Scholar
Bismut, J.-M., Hypoelliptic Laplacian and orbital integrals, Annals of Mathematics Studies, vol. 177 (Princeton University Press, Princeton, NJ, 2011).Google Scholar
Bismut, J.-M. and Labourie, F., Symplectic geometry and the Verlinde formulas, in Surveys in differential geometry: differential geometry inspired by string theory, Surveys in Differential Geometry, vol. 5 (International Press, Boston, MA, 1999), 97311.Google Scholar
Bismut, J.-M. and Shen, S., Intégrales orbitales semi-simples et centre de l'algèbre enveloppante, C. R. Math. Acad. Sci. Paris 357 (2019), 897906.CrossRefGoogle Scholar
Bröcker, T. and tom Dieck, T., Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98 (Springer, New York, 1995). Translated from the German manuscript, Corrected reprint of the 1985 translation.Google Scholar
Chazarain, J. and Piriou, A., Introduction à la théorie des équations aux dérivées partielles linéaires (Gauthier-Villars, Paris, 1981).Google Scholar
Duflo, M., Caractères des groupes et des algèbres de Lie résolubles, Ann. Sci. École Norm. Sup. (4) 3 (1970), 2374.CrossRefGoogle Scholar
Eberlein, P. B., Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1996).Google Scholar
Gelfand, I. M., Graev, M. I. and Pyatetskii-Shapiro, I. I., Representation theory and automorphic functions, Generalized Functions, vol. 6 (Academic Press, Boston, MA, 1990). Translated from the Russian by K. A. Hirsch, Reprint of the 1969 edition.Google Scholar
Harish-Chandra, , The characters of semisimple Lie groups, Trans. Amer. Math. Soc. 83 (1956), 98163.CrossRefGoogle Scholar
Harish-Chandra, , Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87120.CrossRefGoogle Scholar
Harish-Chandra, , A formula for semisimple Lie groups, Amer. J. Math. 79 (1957), 733760.CrossRefGoogle Scholar
Harish-Chandra, , Some results on an invariant integral on a semisimple Lie algebra, Ann. of Math. (2) 80 (1964), 551593.CrossRefGoogle Scholar
Harish-Chandra, , Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119 (1965), 457508.CrossRefGoogle Scholar
Harish-Chandra, , Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1111.CrossRefGoogle Scholar
Harish-Chandra, , Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Funct. Anal. 19 (1975), 104204.CrossRefGoogle Scholar
Hirzebruch, F., Automorphe Formen und der Satz von Riemann-Roch, in Symposium internacional de topología algebraica international symposium on algebraic topology (Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958), 129144.Google Scholar
Hörmander, L., Distribution theory and Fourier analysis, in The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256 (Springer, Berlin, 1983).Google Scholar
Hörmander, L., The analysis of linear partial differential operators. III. Pseudo-differential operators, Grundlehren der Mathematischen Wissenschaften Band 274 (Springer, Berlin, 1985).Google Scholar
Huang, J.-S. and Pandžić, P., Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185202.CrossRefGoogle Scholar
Kawasaki, T., The Riemann–Roch theorem for complex $V$-manifolds, Osaka J. Math. 16 (1979), 151159.Google Scholar
Knapp, A. W., Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36 (Princeton University Press, Princeton, NJ, 1986). An overview based on examples.CrossRefGoogle Scholar
Knapp, A. W., Lie groups beyond an introduction, second edition, Progress in Mathematics, vol. 140 (Birkhäuser, Boston, MA, 2002).Google Scholar
Kostant, B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413455.CrossRefGoogle Scholar
Kostant, B., On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, Adv. Math. 20 (1976), 179212.CrossRefGoogle Scholar
McKean, H. P. and Singer, I. M. Jr., Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 4369.CrossRefGoogle Scholar
Rossmann, W., Kirillov's character formula for reductive Lie groups, Invent. Math. 48 (1978), 207220.CrossRefGoogle Scholar
Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 4787.Google Scholar
Taylor, M. E., Pseudodifferential operators, Princeton Mathematical Series, vol. 34 (Princeton University Press, Princeton, NJ, 1981).CrossRefGoogle Scholar
Varadarajan, V. S., Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, vol. 576 (Springer, Berlin–New York, 1977).CrossRefGoogle Scholar
Vergne, M., On Rossmann's character formula for discrete series, Invent. Math. 54 (1979), 1114.CrossRefGoogle Scholar
Wallach, N. R., Real reductive groups. I, Pure and Applied Mathematics, vol. 132 (Academic Press, Boston, MA, 1988).Google Scholar