Spherical representations and mixed symmetric spaces

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

doi:10.1090/S1088-4165-97-00035-6

Spherical representations and mixed symmetric spaces
HTML articles powered by AMS MathViewer

by Bernhard Krötz, Karl-Hermann Neeb and Gestur Ólafsson PDF

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

Abstract:

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 semi group $\Gamma$. 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.

References

N. Bourbaki, Groupes et algèbres de Lie, Chapitres 7 et 8, Masson, Paris, 1990.
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). MR 1303888, DOI 10.5802/aif.1421
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. MR 514561
J. Hilgert and K.-H. Neeb, Lie–Gruppen und Lie–Algebren, Vieweg Verlag, 1991.
Joachim Hilgert and Karl-Hermann Neeb, Lie semigroups and their applications, Lecture Notes in Mathematics, vol. 1552, Springer-Verlag, Berlin, 1993. MR 1317811, DOI 10.1007/BFb0084640
Joachim Hilgert and Karl-Hermann Neeb, Compression semigroups of open orbits in complex manifolds, Ark. Mat. 33 (1995), no. 2, 293–322. MR 1373026, DOI 10.1007/BF02559711
—, Spherical functions on Ol’shanskiĭ space, J. Funct. Anal., 142 (1996), 446–493.
—, Structure Groups of Euclidian Jordan Algebras and their Representations, in preparation.
J. Hilgert and G. Ólafsson, Causal Symmetric Spaces, Geometry and Harmonic Analysis, Perspectives in Mathematics 18, Academic Press, 1997.
J. Hilgert, G. Ólafsson, and B. Ørsted, Hardy spaces on affine symmetric spaces, J. Reine Angew. Math. 415 (1991), 189–218. MR 1096906, DOI 10.1515/crll.1991.415.189
B. Krötz, On Hardy and Bergman spaces on complex Ol’shanskiĭ semigroups, submitted.
Bernhard Krötz and Karl-Hermann Neeb, On hyperbolic cones and mixed symmetric spaces, J. Lie Theory 6 (1996), no. 1, 69–146. MR 1406006
Jimmie D. Lawson, Polar and Ol′shanskiĭ decompositions, J. Reine Angew. Math. 448 (1994), 191–219. MR 1266749, DOI 10.1515/crll.1994.448.191
Ottmar Loos, Symmetric spaces. I: General theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0239005
Karl-Hermann Neeb, The classification of Lie algebras with invariant cones, J. Lie Theory 4 (1994), no. 2, 139–183. MR 1337190
Karl-Hermann Neeb, Holomorphic representation theory. II, Acta Math. 173 (1994), no. 1, 103–133. MR 1294671, DOI 10.1007/BF02392570
Karl-Hermann Neeb, Holomorphic representations of Ol′shanskiĭ semigroups, Semigroups in algebra, geometry and analysis (Oberwolfach, 1993) De Gruyter Exp. Math., vol. 20, de Gruyter, Berlin, 1995, pp. 241–271. MR 1350335
Karl-Hermann Neeb, Invariant convex sets and functions in Lie algebras, Semigroup Forum 53 (1996), no. 2, 230–261. MR 1400650, DOI 10.1007/BF02574139
—, On the complex and convex geometry of Ol’shanskiĭ semigroups, Institut Mittag-Leffler, 1995-96, preprint.
—, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, de Gruyter, in preparation.
G. Ólafsson and B. Ørsted, The holomorphic discrete series for affine symmetric spaces. I, J. Funct. Anal. 81 (1988), no. 1, 126–159. MR 967894, DOI 10.1016/0022-1236(88)90115-2