Abstract
Let G be a locally compact group and H a closed subgroup of G. We are going to show that there exists a G-invariant Borel measure on the quotient space G∕H, if and only if for the modular functions Δ H and Δ G we have that
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References
Arnal, D., Fujiwara, H., Ludwig, J.: Opérateurs d’entrelacement pour les groupes de Lie exponentiels. Am. J. Math. 118, 839–878 (1996)
Arsac, G.: Opérateurs compacts dans l’espace d’une représentation. C. R. Acad. Sci. Paris 286, 189–192 (1982)
Auslander, L., Kostant, B.: Polarization and unitary representations of solvable Lie groups. Invent. Math. 14, 255–354 (1971)
Auslander, L., Moore, C.C.: Unitary representations of solvable Lie groups. Mem. Am. Math. Soc. 62 pp. 199 (1966)
Baklouti, A., Fujiwara, H.: Opérateurs différentiels associés à certaines représentations unitaires d’un groupe de Lie résoluble exponentiel. Compos. Math. 139, 29–65 (2003)
Baklouti, A., Fujiwara, H.: Commutativité des opérateurs différentiels sur l’espace des représentations restreintes d’un groupe de Lie nilpotent. J. Math. Pures Appl. 83, 137–161 (2004)
Baklouti, A., Fujiwara, H., Ludwig, J.: Analysis of restrictions of unitary representations of a nilpotent Lie group. Bull. Sci. Math. 129, 187–209 (2005)
Benoist, Y.: Espaces symétriques exponentiels. Thèse de 3e cycle, Univ. Paris VII (1983)
Benoist, Y.: Multiplicité un pour les espaces symétriques exponentiels. Mém. Soc. Math. France 15, 1–37 (1984)
Bernat, P., et al.: Représentations des groupes de Lie résolubles. Dunod, Paris (1972)
Blattner, R.J.: On induced representations I. Am. J. Math. 83, 79–98 (1961); Blattner, R.J.: On induced representations II. Am. J. Math. 83, 499–512 (1961)
Bonnet, P.: Transformation de Fourier des distributions de type posotif sur un groupe de Lie unimodulaire. J. Funct. Anal. 55, 220–246 (1984)
Bourbaki, N.: Intégration. Hermann, Paris (1967)
Corwin, L., Greenleaf, F.P., Grélaud, G.: Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups. Trans. Am. Math. Soc. 304, 549–583 (1987)
Corwin, L., Greenleaf, F.P.: Spectrum and multiplicities for restrictions of unitary representations in nilpotent Lie groups. Pac. J. Math. 135, 233–267 (1988)
Corwin, L., Greenleaf, F.P.: Representations of Nilpotent Lie Groups and Their Applications, Part I: Basic Theory and Examples. Cambridge University Press, Cambridge (1990)
Corwin, L., Greenleaf, F.P.: Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity. Commun. Pure Appl. Math. 45, 681–748 (1992)
Dixmier, J.: L’application exponentielle dans les groupes de Lie résolubles. Bull. Soc. Math. France 85, 113–121 (1957)
Dixmier, J.: Représentations irréductibles des algèbres de Lie nilpotentes. Anals Acad. Brasil. Ci. 35, 491–519 (1963)
Dixmier, J.: Les C*-algèbres et leurs représentations. Gauthier-Villars, Paris (1964)
Dixmier, J.: Enveloping algebras. Am. Math. Soc. Graduate Studies in Mathematics, vol. 11 (1996)
Duflo, M.: Open problems in representation theory of Lie groups. In: Oshima, T. (ed.) Conference on Analysis on Homogrneous Spaces, pp 1–5. Katata in Japan (1986)
Effros, E.G.: Transformation groups and C*-algebras. Ann. Math. 81, 38–55 (1965)
Fell, J.M.G.: The dual space of C*-algebras. Trans. Am. Math. Soc. 94, 365–403 (1960)
Fell, J.M.G.: A new proof that nilpotent groups are CCR. Proc. Am. Math. Soc. 107, 93–99 (1962)
Fell, J.M.G.: Weak containment and induced representations of groups I. Can. J. Math. 14, 237–268 (1962); Fell, J.M.G.: Weak containment and induced representations of groups II. Trans. Am. Math. Soc. 110, 424–447 (1964)
Fujiwara, H.: On unitary representations of exponential groups. J. Fac. Sci. Univ. Tokyo 21, 465–471 (1974)
Fujiwara, H.: On holomorphically induced representations of exponential groups. Jpn. J. Math. 4, 109–170 (1978)
Fujiwara, H.: Polarisations réelles et représentations associées d’un groupe de Lie résoluble. J. Funct. Anal. 60, 102–125 (1985)
Fujiwara, H.: Représentations monomiales des groupes de Lie nilpotents. Pac. J. Math. 127, 329–351 (1987)
Fujiwara, H., Yamagami, S.: Certaines représentations monomiales d’un groupe de Lie résoluble exponentiel. Adv. St. Pure Math. 14, 153–190 (1988)
Fujiwara, H.: Représentations monomiales des groupes de Lie résolubles exponentiels. In: Duflo, M., Pedersen, N.V., Vergne, M. (eds.) The Orbit Method in Representation Theory. Proceedings of a Conference in Copenhagen, pp. 61–84. Birkhäser, Boston (1990)
Fujiwara, H.: La formule de Plancherel pour les représentations monomiales des groupes de Lie nilpotents. In: Kawazoe, T., Oshima, T., Sano, S. (eds.) Representation Theory of Lie Groups and Lie Algebras. The Proceedings of Fuji-Kawaguchiko Conference, pp. 140–150. World Scientific (1992)
Fujiwara, H.: Sur les restrictions des représentations unitaires des groupes de Lie résolubles exponentiels. Invent. Math. 104, 647–654 (1991)
Fujiwara, H., Lion, G., Mehdi, S.: On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces. Trans. Am. Math. Soc. 353, 4203–4217 (2001)
Fujiwara, H., Lion, G., Magneron, B.: Algèbres de fonctions associées aux représentations monomiales des groupes de Lie nilpotents. Prépub. Math. Univ. Paris 13, 2002–2 (2002)
Fujiwara, H., Lion, G., Magneron, B., Mehdi, S.: Commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces. Math. Ann. 327, 513–544 (2003)
Fujiwara, H.: Une réciprocité de Frobenius. In: Heyer, H., et al. (eds.) Infinite Dimensional Harmonic Analysis, pp. 17–35. World Scientific (2005)
Fujiwara, H.: Unitary representations of exponential solvable Lie groups–Orbit method (Japanese). Sugakushobo Co. (2010)
Gindikin, G., Pjatetskii-Shapiro, I.I., Vinberg, E.E.: Geometry of homogeneous bounded domains. C. I. M. E. 3, 3–87 (1967)
Godement, R.: Les fonctions de type positif et la théorie des groupes. Trans. Am. Math. Soc. 63, 1–84 (1948)
Grélaud, G.: Désintégration des représentations induites des groupes de Lie résolubles exponentiels. Thèse de 3e cycle, Univ. de Poitiers (1973)
Grélaud, G.: La formule de Plancherel pour les espaces homogènes des groupes de Heisenberg. J. Reine Angew. Math. 398, 92–100 (1989)
Halmos, P.R.: Measure Theory. D. Van Nostrand Company, Inc., New York (1950)
Hochschild, G.: The Structure of Lie Groups. Holden-Day, San Francisco (1965)
Howe, R.: On a connection between nilpotent groups and oscillatory integrals associated to singularities. Pac. J. Math. 73, 329–364 (1977)
Humphreys, J.: Linear algebraic groups. In: Graduate Texts in Mathematics, vol. 21. Springer, New York (1975)
Kirillov, A.A.: Unitary representations of nilpotent Lie groups. Russ. Math. Surv. 17, 57–110 (1962)
Kirillov, A.A.: Elements of the Theory of Representations. Springer, Berlin (1976)
Kirillov, A.A.: Lectures on the orbit method. Am. Math. Soc. Graduate Studies in Mathematics, vol. 64 (2004)
Leptin, H., Ludwig, J.: Unitary Representation Theory of Exponential Lie Groups. Walter de Gruyter, Berlin (1994)
Lion, G.: Indice de Maslov et représentation de Weil. Pub. Math. Univ. Paris VII 2, 45–79 (1978)
Lion, G., Vergne, M.: The Weil Representation, Maslov Index and Theta Series. Birkhäuser, Boston (1980)
Lipsman, R.: Attributes and applications of the Corwin–Greenleaf multiplicity function. Contemp. Math. 177, 27–46 (1994)
Mackey, G.W.: Induced representations of locally compact groups I. Ann. Math. 55, 101–139 (1952); Mackey, G.W.: Induced representations of locally compact groups II. Ann. Math. 58, 193–221 (1953)
Mackey, G.W.: The theory of unitary group representations. Chicago Lectures in Math. (1976)
Moore, C.C.: Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds. Proc. Symp. Pure Math. 26, 3–44 (1973)
Pedersen, N.: On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. Bull. Soc. Math. France 112, 423–467 (1984)
Penney, R.: Abstract Plancherel theorem and a Frobenius reciprocity theorem. J. Funct. Anal. 18, 177–190 (1975)
Pjatetskii-Shapiro, I.I.: Geometry of Classical Domains and Theory of Automorphic Functions. Gordon and Breach, New York (1969)
Pjatetskii-Shapiro, I.I.: On bounded homogeneous domains in an n-dimensional complex space. Izv. Acad. Nauk 26, 107–124 (1962)
Pontrjagin, L.: Topological Groups. Gordon & Breach Science Publishers (1966)
Poulsen, N.S.: On C ∞-vectors and intertwining bilinear forms for representations of Lie groups. J. Funct. Anal. 9, 87–120 (1972)
Pukanszky, L.: Leçon sur les représentations des groupes. Dunod, Paris (1967)
Pukanszky, L.: Representations of solvable Lie groups. Ann. Sci. École Norm. Sup. 4, 464–608 (1971)
Raïs, M.: Représentations des groupes de Lie nilpotents et méthode des orbites, Chap 5. In: d’Analyse Harmonique. Cours du CIMPA (1983)
Rossi, H.: Lectures on representations of groups of holomorphic transformations of Siegel domains. Lecture Notes, Brandeis University (1972)
Rossi, H., Vergne, M.: Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group. J. Funct. Anal. 13, 324–389 (1973)
Saito, M.: Sur certains groupes de Lie résolubles, I. Sci. Papers College Gen. Ed. Univ. Tokyo 7, 1–11 (1957); Saito, M.: Sur certains groupes de Lie résolubles, II. Sci. Papers College Gen. Ed. Univ. Tokyo 7, 157–168 (1957)
Satake, I.: Talks on Lie Groups (Japanese). Nihonhyôronsha (1993)
Satake, I.: Talks on Lie algebras (Japanese). Nihonhyôronsha (2002)
Spanier, H.: Algebraic Topology. McGraw-Hill, New York (1966)
Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs (1964)/McGraw-Hill, New York (1966)
Sugiura, M.: Theory of Lie groups (Japanese). Kyôritu Pub. Co. (2001)
Takenouchi, O.: Sur la facteur-représentation des groupes de Lie de type (E). Math. J. Okayama Univ. 7, 151–161 (1957)
Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic, New York (1967)
Vergne, M.: Étude de certaines représentations induites d’un groupe de Lie résoluble exponentiel. Ann. Sci. Éc. Norm. Sup. 3, 353–384 (1970)
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Fujiwara, H., Ludwig, J. (2015). Induced Representations. In: Harmonic Analysis on Exponential Solvable Lie Groups. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55288-8_3
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