Unitary highest weight representations via the orbit method I: The scalar case

Acta Applicandae Mathematicae

Volumn 44, Issue 1-2, 1996, Pages 151-184

  Hilgert, Joachim ^a   Neeb, Karl Hermann ^b   Ørsted, Bent ^c  

^a CLAUSTHAL UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^c UNIVERSITY OF SOUTHERN DENMARK   (Denmark)

Author keywords

Laplace transform; Nilpotent orbits; Reproducing kernels; Siegel domains

Indexed keywords

EID: 0010794879     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00116520     Document Type: Article

References (25)

1. Bernat, P. et al.: Représentations des groupes de Lie résolubles, Monographies Soc. Math. France, Dunod, Paris, 1972.
2. Bourbaki, N.: Groupes et algèbres de Lie, Chapitres 7-9, Masson, Paris, 1990.
3. Davidson, M. G., Enright, T. J., and Stanke, R. J.: Differential operators and highest weight representations, Mem. Amer. Math. Soc. 455 (1991).
4. Enright, T. J., Howe, R., and Wallach, N.: A classification of unitary highest weight modules, in Proc. Representation Theory of Reductive Groups, Park City, UT, 1982, Progr. Math. 40, Birkhäuser Verlag, Basel, 1983, pp. 97-149.
5. Enright, T. J. and Joseph, A.: An intrinsic classification of unitarizable highest weight modules, Math. Ann. 288 (1990), 571-594.
6. Faraut, J. and Koranyi, A.: Analysis on Symmetric Cones, Oxford Math. Monographs, Oxford University Press, 1994.
7. Guillemin, V. and Sternberg, S.: Geometric Asymptotics, AMS, Providence, 1977.
8. Harish-Chandra: Representations of semi-simple Lie groups, VI, Amer. J. Math. 78 (1956), 564-628.
9. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, London, 1978.
10. Hilgert, J., Hofmann, K. H., and Lawson, J. D.: Lie Groups, Convex Cones, and Semigroups, Clarendon Press, Oxford, 1989.
11. Hilgert, J., Neeb, K.-H., and Ørsted, B.: The geometry of nilpotent coadjoint orbits of convex type in Hermitian Lie algebras, J. Lie Theory 4(2) (1994), 47-97.
12. Hilgert, J., Neeb, K.-H., and Ørsted, B.: Conal Heisenberg algebras and associated Hilbert spaces, J. Reine angew. Math., to appear.
13. Jakobsen, H. P.: Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385-412.
14. Jakobsen, H. P.: An intrinsic classification of the unitarizable highest weight modules as well as their associated varieties, Preprint, 1992.
15. Knapp, A. W. and Okamoto, K.: Limits of holomorphic discrete series, J. Funct. Anal. 9 (1972), 375-409.
16. Neeb, K.-H.: Coherent states, holomorphic extensions, and highest weight representations, Pacific J. Math., to appear.
17. Neeb, K.-H.: Realization of general unitary highest weight representations, Preprint, Technische Hochschule Darmstadt 1662, 1994.
18. Neeb, K.-H.: Holomorphic representation theory II, Acta Math. 173(1) (1994), 103-133.
19. Neeb, K.-H.: Semi-group actions on reproducing kernel spaces, Submitted.
20. Rawnsley, J., Schmid, W., and Wolf, J.: Singular unitary representations and indefinite harmonic theory, J. Funct. Anal. 51 (1983), 1-114.
21. Satake, I.: Algebraic Structures of Symmetric Domains, Publ. Math. Soc. Japan 14, Princeton Univ. Press, 1980.
22. Vergne, M. and Rossi, H.: Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1976), 1-59.
23. Wallach, N.: On the unitarizability of representations with highest weight, in Non-commutative Harmonic Analysis, Lecture Notes in Math. 466, Springer, New York, 1975, pp. 226-231.
24. Warner, G.: Harmonic Analysis on Semi-simple Lie Groups 1, Springer, Berlin, 1972.
25. Weil, A.: Variétés Kähleriennes, Hermann, Paris, 1958.