To a pair of a Lie group and an open elliptic convex cone in its Lie algebra one associates a complex semigroup which permits an action of by biholomorphic mappings. In the case where is a vector space is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain is Stein is and only if it is of the form , with convex, that each holomorphic function on extends to the smallest biinvariant Stein domain containing , and that biinvariant plurisubharmonic functions on correspond to invariant convex functions on .
À tout cône ouvert elliptique convexe dans l’algèbre de Lie d’un groupe de Lie on associe un semi-groupe complexe qui permet une action holomorphe de . Si est l’algèbre de Lie toute entière, le semi-groupe est un groupe complexe réductif. Dans cet article on montre que chaque semi-groupe est une variété de Stein, qu’un domaine biinvariant est de Stein si et seulement si où est convexe, que toute fonction holomorphe sur s’étend au plus petit domaine de Stein contenant , et que les fonctions biinvariantes plurisousharmoniques sur correspondent aux fonctions convexes sur .
@article{AIF_1998__48_1_149_0, author = {Neeb, Karl-Hermann}, title = {On the complex and convex geometry of {Ol'shanskii} semigroups}, journal = {Annales de l'Institut Fourier}, pages = {149--203}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {48}, number = {1}, year = {1998}, doi = {10.5802/aif.1614}, mrnumber = {99e:22013}, zbl = {0901.22003}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1614/} }
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 SP - 149 EP - 203 VL - 48 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1614/ DO - 10.5802/aif.1614 LA - en ID - AIF_1998__48_1_149_0 ER -
%0 Journal Article %A Neeb, Karl-Hermann %T On the complex and convex geometry of Ol'shanskii semigroups %J Annales de l'Institut Fourier %D 1998 %P 149-203 %V 48 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1614/ %R 10.5802/aif.1614 %G en %F AIF_1998__48_1_149_0
Neeb, Karl-Hermann. On the complex and convex geometry of Ol'shanskii semigroups. Annales de l'Institut Fourier, Volume 48 (1998) no. 1, pp. 149-203. doi : 10.5802/aif.1614. http://archive.numdam.org/articles/10.5802/aif.1614/
[AL92] Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. N. S., 3(4) (1992), 365-375. | MR | Zbl
and ,[Fe94] Differentialgeometrische Charakterisierung invarianter Holomorphiegebiete, Schriftenreihe des Graduiertenkollegs “Geometrische und mathematische Physik”, Universität Bochum, 7, 1994. | Zbl
,[He93] Equivariant holomorphic extensions of real analytic manifolds, Bull. Soc. Math. France, 121 (1993), 445-463. | EuDML | Numdam | MR | Zbl
,[HHL89] “Lie Groups, Convex Cones, and Semigroups”, Oxford University Press, 1989. | MR | Zbl
, and ,[HiNe93] “Lie semigroups and their applications”, Lecture Notes in Math., 1552, Springer, 1993. | Zbl
and ,[HNP94] Symplectic convexity theorems and coadjoint orbits, Comp. Math., 94 (1994), 129-180. | EuDML | Numdam | MR | Zbl
, and ,[Hö73] An introduction to complex analysis in several variables, North-Holland, 1973. | Zbl
,[Las78] Sur la transformation de Fourier-Laurent dans un groupe analytique complexe réductif, Ann. Inst. Fourier, Grenoble, 28-1 (1978), 115-138. | EuDML | Numdam | MR | Zbl
,[MaMo60] “Sur certains espaces fibrés holomorphes sur une variété de Stein”, Bull. Soc. Math. France, 88 (1960), 137-155. | EuDML | Numdam | MR | Zbl
, and ,[Ne94a] Holomorphic representation theory II, Acta Math., 173-1 (1994), 103-133. | MR | Zbl
,[Ne94b] Realization of general unitary highest weight representations, Preprint 1662, Technische Hochschule Darmstadts, 1994.
,[Ne94c] 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. | Zbl
,[Ne94d] A convexity theorem for semisimple symmetric spaces, Pacific Journal of Math., 162-2 (1994), 305-349. | MR | Zbl
,[Ne94e] On closedness and simple connectedness of adjoint and coadjoint orbits, Manuscripta Math., 82 (1994), 51-65. | MR | Zbl
,[Ne95a] Holomorphic representation theory I, Math., Ann., 301 (1995), 155-181. | MR | Zbl
,[Ne95b] Holomorphic representations of Ol'shanskiĠ semigroups, in “Semigroups in Algebra, Geometry and Analysis”, K. H. Hofmann et al., eds., de Gruyter, 1995. | MR | Zbl
,[Ne96a] Invariant Convex Sets and Functions in Lie Algebras, Semigroup Forum 53 (1996), 230-261. | MR | Zbl
,[Ne96b] Coherent states, holomorphic extensions, and highest weight representations, Pac. J. Math., 174-2 (1996), 497-542. | Zbl
,[Ne98] “Holomorphy and Convexity in Lie Theory”, de Gruyter, Expositions in Mathematics, to appear. | Zbl
,[Ra86] Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, New York, 1986. | MR | Zbl
,[Ro63] On Envelopes of Holomorphy, Comm. on Pure and Appl. Math., 16 (1963), 9-17. | MR | Zbl
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