Spherical representations and mixed symmetric spaces. (English) Zbl 0887.22022

Krötz, Bernhard; Neeb, Karl-Hermann; Ólafsson, Gestur

Represent. Theory 1, 424-461 (1997).

Summary: Let \(G/H\) be a symmetric space admitting a \(G\)-invariant hyperbolic cone field. For each such cone field we construct a local tube domain \(\Xi\) containing \(G/H\) as a boundary component. The domain \(\Xi\) is an orbit of an Ol’shanskii type semigroup \(\Gamma\). We describe the structure of the group \(G\) and the domain \(\Xi\). Furthermore we explore the correspondence between \(\Gamma\)-modules of holomorphic sections of line bundles over \(\Xi\) and spherical highest weight modules.

MSC:

22E47	Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E15	General properties and structure of real Lie groups
53C35	Differential geometry of symmetric spaces
54H15	Transformation groups and semigroups (topological aspects)

Keywords:

spherical representations; symmetric space; hyperbolic cone field; orbit; Ol’shanskii type semigroup; spherical highest weight modules

References:

[1]	N. Bourbaki, Groupes et algèbres de Lie, Chapitres 7 et 8, Masson, Paris, 1990. · Zbl 0199.35203
[2]	J. Faraut, J. Hilgert, and G. Ólafsson, Spherical functions on ordered symmetric spaces, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 3, 927 – 965 (English, with English and French summaries). · Zbl 0810.43003
[3]	Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0451.53038
[4]	J. Hilgert and K.-H. Neeb, Lie-Gruppen und Lie-Algebren, Vieweg Verlag, 1991. · Zbl 0760.22005
[5]	Joachim Hilgert and Karl-Hermann Neeb, Lie semigroups and their applications, Lecture Notes in Mathematics, vol. 1552, Springer-Verlag, Berlin, 1993. · Zbl 0807.22001
[6]	Joachim Hilgert and Karl-Hermann Neeb, Compression semigroups of open orbits in complex manifolds, Ark. Mat. 33 (1995), no. 2, 293 – 322. · Zbl 0855.22010 · doi:10.1007/BF02559711
[7]	-, Spherical functions on Ol’shanskii space, J. Funct. Anal., 142 (1996), 446-493. CMP 97:05
[8]	-, Structure Groups of Euclidian Jordan Algebras and their Representations, in preparation.
[9]	J. Hilgert and G. Ólafsson, Causal Symmetric Spaces, Geometry and Harmonic Analysis, Perspectives in Mathematics 18, Academic Press, 1997. CMP 96:17 · Zbl 0931.53004
[10]	J. Hilgert, G. Ólafsson, and B. Ørsted, Hardy spaces on affine symmetric spaces, J. Reine Angew. Math. 415 (1991), 189 – 218. · Zbl 0716.43006
[11]	B. Krötz, On Hardy and Bergman spaces on complex Ol’shanskii semigroups, submitted.
[12]	Bernhard Krötz and Karl-Hermann Neeb, On hyperbolic cones and mixed symmetric spaces, J. Lie Theory 6 (1996), no. 1, 69 – 146. · Zbl 0860.22004
[13]	Jimmie D. Lawson, Polar and Ol\(^{\prime}\)shanskiĭ decompositions, J. Reine Angew. Math. 448 (1994), 191 – 219. · Zbl 0786.22012 · doi:10.1515/crll.1994.448.191
[14]	Ottmar Loos, Symmetric spaces. I: General theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Ottmar Loos, Symmetric spaces. II: Compact spaces and classification, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
[15]	Karl-Hermann Neeb, The classification of Lie algebras with invariant cones, J. Lie Theory 4 (1994), no. 2, 139 – 183. · Zbl 0838.17003
[16]	Karl-Hermann Neeb, Holomorphic representation theory. II, Acta Math. 173 (1994), no. 1, 103 – 133. · Zbl 0842.22004 · doi:10.1007/BF02392570
[17]	Karl-Hermann Neeb, Holomorphic representations of Ol\(^{\prime}\)shanskiĭ semigroups, Semigroups in algebra, geometry and analysis (Oberwolfach, 1993) De Gruyter Exp. Math., vol. 20, de Gruyter, Berlin, 1995, pp. 241 – 271. · Zbl 0851.22014
[18]	Karl-Hermann Neeb, Invariant convex sets and functions in Lie algebras, Semigroup Forum 53 (1996), no. 2, 230 – 261. · Zbl 0873.17009 · doi:10.1007/BF02574139
[19]	-, On the complex and convex geometry of Ol’shanskii semigroups, Institut Mittag-Leffler, 1995-96, preprint.
[20]	-, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, de Gruyter, in preparation. · Zbl 0936.22001
[21]	G. Ólafsson and B. Ørsted, The holomorphic discrete series for affine symmetric spaces. I, J. Funct. Anal. 81 (1988), no. 1, 126 – 159. · Zbl 0678.22008 · doi:10.1016/0022-1236(88)90115-2