On the complex and convex geometry of Ol'shanskii semigroups

Karl-Hermann Neeb

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 1, page 149-203
  • ISSN: 0373-0956

Abstract

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To a pair of a Lie group G and an open elliptic convex cone W in its Lie algebra one associates a complex semigroup S = G Exp ( i W ) which permits an action of G × G by biholomorphic mappings. In the case where W is a vector space S is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain D ⊆ S is Stein is and only if it is of the form G Exp ( D h ) , with D h ⊆ i W convex, that each holomorphic function on D extends to the smallest biinvariant Stein domain containing D , and that biinvariant plurisubharmonic functions on D correspond to invariant convex functions on D h .

How to cite

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Neeb, Karl-Hermann. "On the complex and convex geometry of Ol'shanskii semigroups." Annales de l'institut Fourier 48.1 (1998): 149-203. <http://eudml.org/doc/75274>.

@article{Neeb1998,
abstract = {To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G\{\rm Exp\}(iW)$ which permits an action of $G\times G$ by biholomorphic mappings. In the case where $W$ is a vector space $S$ is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain $D\subseteq S$ is Stein is and only if it is of the form $G\{\rm Exp\}(D_h)$, with $Dh\subseteq iW$ convex, that each holomorphic function on $D$ extends to the smallest biinvariant Stein domain containing $D$, and that biinvariant plurisubharmonic functions on $D$ correspond to invariant convex functions on $D_h$.},
author = {Neeb, Karl-Hermann},
journal = {Annales de l'institut Fourier},
keywords = {Ol'shanskii semigroup; classical Lie group; real Lie group; complex reductive Lie group; Lie algebras},
language = {eng},
number = {1},
pages = {149-203},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the complex and convex geometry of Ol'shanskii semigroups},
url = {http://eudml.org/doc/75274},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Neeb, Karl-Hermann
TI - On the complex and convex geometry of Ol'shanskii semigroups
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 1
SP - 149
EP - 203
AB - To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G{\rm Exp}(iW)$ which permits an action of $G\times G$ by biholomorphic mappings. In the case where $W$ is a vector space $S$ is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain $D\subseteq S$ is Stein is and only if it is of the form $G{\rm Exp}(D_h)$, with $Dh\subseteq iW$ convex, that each holomorphic function on $D$ extends to the smallest biinvariant Stein domain containing $D$, and that biinvariant plurisubharmonic functions on $D$ correspond to invariant convex functions on $D_h$.
LA - eng
KW - Ol'shanskii semigroup; classical Lie group; real Lie group; complex reductive Lie group; Lie algebras
UR - http://eudml.org/doc/75274
ER -

References

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  1. [AL92] H. AZAD and J.-J. LOEB, Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. N. S., 3(4) (1992), 365-375. Zbl0777.32008MR94a:32014
  2. [Fe94] G. FELS, Differentialgeometrische Charakterisierung invarianter Holomorphiegebiete, Schriftenreihe des Graduiertenkollegs “Geometrische und mathematische Physik”, Universität Bochum, 7, 1994. Zbl0850.32001
  3. [He93] P. HEINZNER, Equivariant holomorphic extensions of real analytic manifolds, Bull. Soc. Math. France, 121 (1993), 445-463. Zbl0794.32022MR94i:32050
  4. [HHL89] J. HILGERT, K.H. HOFMANN and J.D. LAWSON, “Lie Groups, Convex Cones, and Semigroups”, Oxford University Press, 1989. Zbl0701.22001MR91k:22020
  5. [HiNe93] J. HILGERT and K.-H. NEEB, “Lie semigroups and their applications”, Lecture Notes in Math., 1552, Springer, 1993. Zbl0807.22001
  6. [HNP94] J. HILGERT, K.-H. NEEB and W. PLANK, Symplectic convexity theorems and coadjoint orbits, Comp. Math., 94 (1994), 129-180. Zbl0819.22006MR96d:53053
  7. [Hö73] L. HÖRMANDER, An introduction to complex analysis in several variables, North-Holland, 1973. Zbl0271.32001
  8. [Las78] M. LASALLE, Sur la transformation de Fourier-Laurent dans un groupe analytique complexe réductif, Ann. Inst. Fourier, Grenoble, 28-1 (1978), 115-138. Zbl0334.32028MR80b:32006
  9. [MaMo60] Y. MATSUSHIMA, and A. MORIMOTO, “Sur certains espaces fibrés holomorphes sur une variété de Stein”, Bull. Soc. Math. France, 88 (1960), 137-155. Zbl0094.28104MR23 #A1061
  10. [Ne94a] K.-H. NEEB, Holomorphic representation theory II, Acta Math., 173-1 (1994), 103-133. Zbl0842.22004MR96a:22025
  11. [Ne94b] K.-H. NEEB, Realization of general unitary highest weight representations, Preprint 1662, Technische Hochschule Darmstadts, 1994. 
  12. [Ne94c] K.-H. NEEB, A Duistermaat-Heckman formula for admissible coadjoint orbits, Proceedings of “Workshop on Lie Theory and its applications in Physics”, Clausthal, August, 1995, Eds. Doebner, Dobrev, to appear. Zbl0920.22009
  13. [Ne94d] K.-H. NEEB, A convexity theorem for semisimple symmetric spaces, Pacific Journal of Math., 162-2 (1994), 305-349. Zbl0809.53058MR95b:22016
  14. [Ne94e] K.-H. NEEB, On closedness and simple connectedness of adjoint and coadjoint orbits, Manuscripta Math., 82 (1994), 51-65. Zbl0815.22004MR94k:22012
  15. [Ne95a] K.-H. NEEB, Holomorphic representation theory I, Math., Ann., 301 (1995), 155-181. Zbl0829.43017MR96a:22024
  16. [Ne95b] K.-H. NEEB, Holomorphic representations of Ol'shanskiĠ semigroups, in “Semigroups in Algebra, Geometry and Analysis”, K. H. Hofmann et al., eds., de Gruyter, 1995. Zbl0851.22014MR97b:22017
  17. [Ne96a] K.-H. NEEB, Invariant Convex Sets and Functions in Lie Algebras, Semigroup Forum 53 (1996), 230-261. Zbl0873.17009MR97j:17033
  18. [Ne96b] K.-H. NEEB, Coherent states, holomorphic extensions, and highest weight representations, Pac. J. Math., 174-2 (1996), 497-542. Zbl0894.22008
  19. [Ne98] K.-H. NEEB, “Holomorphy and Convexity in Lie Theory”, de Gruyter, Expositions in Mathematics, to appear. Zbl0936.22001
  20. [Ra86] R. M. RANGE, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, New York, 1986. Zbl0591.32002MR87i:32001
  21. [Ro63] H. ROSSI, On Envelopes of Holomorphy, Comm. on Pure and Appl. Math., 16 (1963), 9-17. Zbl0113.06001MR26 #6436

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