Unitary Representations of Reductive Lie Groups

Vogan, David A.jun.

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 11 (2000), p. 147-167 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

One of the fundamental problems of abstract harmonic analysis is the determination of the irreducible unitary representations of simple Lie groups. After recalling why this problem is of interest, we discuss the present state of knowledge about it. In the language of Kirillov and Kostant, the problem finally is to «quantize» nilpotent coadjoint orbits.

Détail

Publié le : 2000-12-01

MR 1845669 | Zbl 1149.22301

Bibliographie

[1] Arnold, V., Mathematical Methods of Classical Mechanics. Springer-Verlag, New York-Heidelberg-Berlin1978. | MR 690288 | Zbl 0386.70001

[2] Barbasch, D. - Vogan, D., Unipotent representations of complex semisimple Lie groups. Ann. of Math., 121, 1985,41-110. | MR 782556 | Zbl 0582.22007

[3] Borel, A. - Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Princeton University Press, Princeton, New Jersey 1980. | MR 554917

[4] Dixmier, J., Les $C^{\star}$-algèbres et leurs représentations. Gauthier-Villars, Paris1964. | MR 171173

[5] Duflo, M., Théorie de Mackey pour les groupes de Lie algébriques. Acta Math., 149, 1982, 153-213. | MR 688348 | Zbl 0529.22011

[6] Guichardet, A., Cohomologie des Groupes Topologiques et des Algèbres de Lie. CEDIC, Paris1980. | MR 644979

[7] Knapp, A., Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton University Press, Princeton, New Jersey1986. | MR 855239 | Zbl 0604.22001

[8] Kostant, B., Quantization and unitary representations. In: C. Taam (ed.), Lectures in Modem Analysis and Applications. Lecture Notes in Mathematics, 170, Springer-Verlag, Berlin-Heidelberg-New York 1970. | MR 294568

[9] Kumaresan, S., On the canonical k-types in the irreducible unitary g-modules with non-zero relative cohomology. Invent. Math., 59, 1980, 1-11. | MR 575078 | Zbl 0442.22010

[10] Makey, G., Mathematical Foundations of Quantum Mechanics. W. A. Benjamin Inc., New York1963. | MR 155567

[11] Makey, G., Theory of Unitary Group Representations. University of Chicago Press, Chicago1976. | MR 396826

[12] Mcgovern, W., Rings of regular functions on nilpotent orbits and their covers. Inv. Math., 97, 1989, 209-217. | MR 999319 | Zbl 0648.22004

[13] Schmid, W., $L_{2}$ cohomology and the discrete series. Ann. of Math., 103, 1976, 375-394. | MR 396856 | Zbl 0333.22009

[14] Torasso, P., Méthode des orbites de Kirillov-Duflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle. Duke Math. J., 90, 1997, 261-377. | MR 1484858 | Zbl 0941.22017

[15] Vogan, D., Unitarizability of certain series of representations. Ann. of Math., 120, 1984, 141-187. | MR 750719 | Zbl 0561.22010

[16] Vogan, D., Associated varieties and unipotent representations. In: W. Barker - P. Sally (eds.), Harmonic Analysis on Reductive Groups. Birkhäuser, Boston-Basel-Berlin 1991, 315-388. | MR 1168491 | Zbl 0832.22019

[17] Vogan, D., The unitary dual of $G_{2}$. Invent. Math., 116, 1994, 677-791. | MR 1253210 | Zbl 0808.22003

[18] Vogan, D., The method of coadjoint orbits for real reductive groups. Representation Theory of Lie Groups, IAS/Park City Mathematics Series, 8, American Mathematical Society, Providence, RI, 1999. | MR 1737729

[19] Vogan, D. - Zuckerman, G., Unitary representations with non-zero cohomology. Compositio Math., 53, 1984, 51-90. | MR 762307 | Zbl 0692.22008