Abstract
In this work we present a new context of the canonical representations which have been introduced by Berezin, Gel’fand, Graev and Vershik for simple Lie groups G of Hermitian type. We discuss maximal-degenerate representations of the complexification of G and the decomposition of the canonical representations into irreducible parts.
Mathematics Subject Classification (1991): 22E30, 22E46, 43A80, 43A85, 53C35
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berezin, F.A.: Quantization in complex symmetric spaces, Math. USSR Izv. 9 (1975) 341–379.
Van Dijk, G. and Hille, S.C.: Canonical representations related to hyperbolic spaces, Report no. 3, Institut Mittag Leffler, Djursholm, 1995/96.
Van Dijk, G. and Molchanov, V.F.: Tensor products of maximal degenerate representations of SL(n, ℝ), in preparation.
Erdélyi, A., et al.: Tables of Integral Transforms, II, McGraw-Hill, New York, 1954.
Faraut, J. and Koranyi, A.: Function spaces and reproducing kernels on bounded symmetric domains, J. Funct Anal. 88 (1990) 64–89.
Hille, S.C.: Maximal degenerate representations of SL(n + 1,ℂ), Preprint Leiden University, 1996.
Johnson, K.D.: Degenerate principal series representations and tube domains, Contemporary Math. 138 (1992) 175–187.
Molchanov, V.F.: On quantization on para-Hermitian symmetric spaces, In: Adv. in Math. Sciences, Amer. Math. Soc. (1996).
Ørsted, B. and Zhang, G.: Generalized principal series representations and tube domains, Duke Math. J. 78 (1995) 335–358.
Unterberger, A. and Upmeier, H.: The Berezin transform and invariant differential operators, Preprint, 1994.
Vershik, A.M., Gel’fand, I.M. and Graev, M.I.: Representations of the group SL(2, R) where R is a ring of functions, In: Representation Theory. Cambridge University Press, Cambridge, 1982.
Wolf, J.A.: The action of a real semisimple group on a complex flag manifold. I: orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969) 1121–1236.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Van Dijk, G., Hille, S.C. (1998). Maximal Degenerate Representations, Berezin Kernels and Canonical Representations. In: Komrakov, B.P., Krasil’shchik, I.S., Litvinov, G.L., Sossinsky, A.B. (eds) Lie Groups and Lie Algebras. Mathematics and Its Applications, vol 433. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5258-7_18
Download citation
DOI: https://doi.org/10.1007/978-94-011-5258-7_18
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6212-1
Online ISBN: 978-94-011-5258-7
eBook Packages: Springer Book Archive