Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Preliminaries: Lie Groups and Lie Algebras Read chapter Read first chapter

2015 | OriginalPaper | Chapter

2. Haar Measure and Group Algebra

Authors : Hidenori Fujiwara, Jean Ludwig

Published in: Harmonic Analysis on Exponential Solvable Lie Groups

Publisher: Springer Japan

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Excerpt

Let G be a locally compact topological group. We denote by C c (G) the vector space of all the continuous complex-valued functions on G with compact support. On this space we put the norm
$$\displaystyle{ \|f\|_{\infty }:=\sup _{g\in G}\vert f(g)\vert,\quad g \in G. }$$
The completion of C c (G) with respect to this norm is the function-space
$$\displaystyle{ C_{0}(G):=\{ f: G \rightarrow \mathbb{C};f\text{ continuous, tending to }0\text{ at }\infty \}. }$$
The support of a continuous function defined on a topological space X is by definition the closure supp(f) of the subset {x ∈ X;  f(x) ≠ 0}. …

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Preliminaries: Lie Groups and Lie Algebras
next chapter Induced Representations
Literature
1.
Arnal, D., Fujiwara, H., Ludwig, J.: Opérateurs d’entrelacement pour les groupes de Lie exponentiels. Am. J. Math. 118, 839–878 (1996)CrossRefMATHMathSciNet
2.
Arsac, G.: Opérateurs compacts dans l’espace d’une représentation. C. R. Acad. Sci. Paris 286, 189–192 (1982)
3.
4.
Auslander, L., Moore, C.C.: Unitary representations of solvable Lie groups. Mem. Am. Math. Soc. 62 pp. 199 (1966)MathSciNet
5.
Baklouti, A., Fujiwara, H.: Opérateurs différentiels associés à certaines représentations unitaires d’un groupe de Lie résoluble exponentiel. Compos. Math. 139, 29–65 (2003)CrossRefMATHMathSciNet
6.
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)CrossRefMATHMathSciNet
7.
Baklouti, A., Fujiwara, H., Ludwig, J.: Analysis of restrictions of unitary representations of a nilpotent Lie group. Bull. Sci. Math. 129, 187–209 (2005)CrossRefMATHMathSciNet
8.
Benoist, Y.: Espaces symétriques exponentiels. Thèse de 3 e cycle, Univ. Paris VII (1983)
9.
Benoist, Y.: Multiplicité un pour les espaces symétriques exponentiels. Mém. Soc. Math. France 15, 1–37 (1984)MATHMathSciNet
10.
Bernat, P., et al.: Représentations des groupes de Lie résolubles. Dunod, Paris (1972)MATH
11.
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)
12.
Bonnet, P.: Transformation de Fourier des distributions de type posotif sur un groupe de Lie unimodulaire. J. Funct. Anal. 55, 220–246 (1984)CrossRefMATHMathSciNet
13.
Bourbaki, N.: Intégration. Hermann, Paris (1967)
14.
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)CrossRefMATH
15.
Corwin, L., Greenleaf, F.P.: Spectrum and multiplicities for restrictions of unitary representations in nilpotent Lie groups. Pac. J. Math. 135, 233–267 (1988)CrossRefMATHMathSciNet
16.
Corwin, L., Greenleaf, F.P.: Representations of Nilpotent Lie Groups and Their Applications, Part I: Basic Theory and Examples. Cambridge University Press, Cambridge (1990)MATH
17.
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)CrossRefMATHMathSciNet
18.
Dixmier, J.: L’application exponentielle dans les groupes de Lie résolubles. Bull. Soc. Math. France 85, 113–121 (1957)MATHMathSciNet
19.
Dixmier, J.: Représentations irréductibles des algèbres de Lie nilpotentes. Anals Acad. Brasil. Ci. 35, 491–519 (1963)MATHMathSciNet
20.
Dixmier, J.: Les C*-algèbres et leurs représentations. Gauthier-Villars, Paris (1964)
21.
Dixmier, J.: Enveloping algebras. Am. Math. Soc. Graduate Studies in Mathematics, vol. 11 (1996)
22.
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)
23.
24.
25.
26.
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)
27.
Fujiwara, H.: On unitary representations of exponential groups. J. Fac. Sci. Univ. Tokyo 21, 465–471 (1974)MATH
28.
Fujiwara, H.: On holomorphically induced representations of exponential groups. Jpn. J. Math. 4, 109–170 (1978)MATH
29.
Fujiwara, H.: Polarisations réelles et représentations associées d’un groupe de Lie résoluble. J. Funct. Anal. 60, 102–125 (1985)CrossRefMATHMathSciNet
30.
Fujiwara, H.: Représentations monomiales des groupes de Lie nilpotents. Pac. J. Math. 127, 329–351 (1987)MATH
31.
Fujiwara, H., Yamagami, S.: Certaines représentations monomiales d’un groupe de Lie résoluble exponentiel. Adv. St. Pure Math. 14, 153–190 (1988)MathSciNet
32.
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)CrossRef
33.
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)
34.
Fujiwara, H.: Sur les restrictions des représentations unitaires des groupes de Lie résolubles exponentiels. Invent. Math. 104, 647–654 (1991)CrossRefMATHMathSciNet
35.
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)CrossRefMATHMathSciNet
36.
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)
37.
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)CrossRefMATHMathSciNet
38.
Fujiwara, H.: Une réciprocité de Frobenius. In: Heyer, H., et al. (eds.) Infinite Dimensional Harmonic Analysis, pp. 17–35. World Scientific (2005)
39.
Fujiwara, H.: Unitary representations of exponential solvable Lie groups–Orbit method (Japanese). Sugakushobo Co. (2010)
40.
Gindikin, G., Pjatetskii-Shapiro, I.I., Vinberg, E.E.: Geometry of homogeneous bounded domains. C. I. M. E. 3, 3–87 (1967)
41.
Godement, R.: Les fonctions de type positif et la théorie des groupes. Trans. Am. Math. Soc. 63, 1–84 (1948)MATHMathSciNet
42.
Grélaud, G.: Désintégration des représentations induites des groupes de Lie résolubles exponentiels. Thèse de 3 e cycle, Univ. de Poitiers (1973)
43.
Grélaud, G.: La formule de Plancherel pour les espaces homogènes des groupes de Heisenberg. J. Reine Angew. Math. 398, 92–100 (1989)MATHMathSciNet
44.
45.
Hochschild, G.: The Structure of Lie Groups. Holden-Day, San Francisco (1965)MATH
46.
Howe, R.: On a connection between nilpotent groups and oscillatory integrals associated to singularities. Pac. J. Math. 73, 329–364 (1977)CrossRefMATHMathSciNet
47.
Humphreys, J.: Linear algebraic groups. In: Graduate Texts in Mathematics, vol. 21. Springer, New York (1975)
48.
49.
50.
Kirillov, A.A.: Lectures on the orbit method. Am. Math. Soc. Graduate Studies in Mathematics, vol. 64 (2004)
51.
Leptin, H., Ludwig, J.: Unitary Representation Theory of Exponential Lie Groups. Walter de Gruyter, Berlin (1994)CrossRefMATH
52.
Lion, G.: Indice de Maslov et représentation de Weil. Pub. Math. Univ. Paris VII 2, 45–79 (1978)MathSciNet
53.
Lion, G., Vergne, M.: The Weil Representation, Maslov Index and Theta Series. Birkhäuser, Boston (1980)MATH
54.
Lipsman, R.: Attributes and applications of the Corwin–Greenleaf multiplicity function. Contemp. Math. 177, 27–46 (1994)CrossRefMathSciNet
55.
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)
56.
Mackey, G.W.: The theory of unitary group representations. Chicago Lectures in Math. (1976)
57.
Moore, C.C.: Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds. Proc. Symp. Pure Math. 26, 3–44 (1973)CrossRef
58.
Pedersen, N.: On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. Bull. Soc. Math. France 112, 423–467 (1984)MATHMathSciNet
59.
60.
Pjatetskii-Shapiro, I.I.: Geometry of Classical Domains and Theory of Automorphic Functions. Gordon and Breach, New York (1969)
61.
Pjatetskii-Shapiro, I.I.: On bounded homogeneous domains in an n-dimensional complex space. Izv. Acad. Nauk 26, 107–124 (1962)
62.
Pontrjagin, L.: Topological Groups. Gordon   &  Breach Science Publishers (1966)
63.
Poulsen, N.S.: On C ∞ -vectors and intertwining bilinear forms for representations of Lie groups. J. Funct. Anal. 9, 87–120 (1972)CrossRefMATHMathSciNet
64.
Pukanszky, L.: Leçon sur les représentations des groupes. Dunod, Paris (1967)
65.
Pukanszky, L.: Representations of solvable Lie groups. Ann. Sci. École Norm. Sup. 4, 464–608 (1971)MathSciNet
66.
Raïs, M.: Représentations des groupes de Lie nilpotents et méthode des orbites, Chap 5. In: d’Analyse Harmonique. Cours du CIMPA (1983)
67.
Rossi, H.: Lectures on representations of groups of holomorphic transformations of Siegel domains. Lecture Notes, Brandeis University (1972)MATH
68.
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)CrossRefMATHMathSciNet
69.
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)
70.
Satake, I.: Talks on Lie Groups (Japanese). Nihonhyôronsha (1993)
71.
Satake, I.: Talks on Lie algebras (Japanese). Nihonhyôronsha (2002)
72.
Spanier, H.: Algebraic Topology. McGraw-Hill, New York (1966)MATH
73.
Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs (1964)/McGraw-Hill, New York (1966)
74.
Sugiura, M.: Theory of Lie groups (Japanese). Kyôritu Pub. Co. (2001)
75.
Takenouchi, O.: Sur la facteur-représentation des groupes de Lie de type (E). Math. J. Okayama Univ. 7, 151–161 (1957)MATHMathSciNet
76.
Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic, New York (1967)MATH
77.
Vergne, M.: Étude de certaines représentations induites d’un groupe de Lie résoluble exponentiel. Ann. Sci. Éc. Norm. Sup. 3, 353–384 (1970)MATHMathSciNet
Metadata
Title
Haar Measure and Group Algebra
Authors
Hidenori Fujiwara
Jean Ludwig
Copyright Year
2015
Publisher
Springer Japan
Book
Harmonic Analysis on Exponential Solvable Lie Groups

Print ISBN: 978-4-431-55287-1

Electronic ISBN: 978-4-431-55288-8

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-4-431-55288-8_2

Premium Partner

    Image Credits