On the complex and convex geometry of Ol'shanskii semigroups
Annales de l'Institut Fourier, Volume 48 (1998) no. 1, pp. 149-203.

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 (iW) 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 DS is Stein is and only if it is of the form G Exp (D h ), with DhiW 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 .

À tout cône ouvert elliptique convexe W dans l’algèbre de Lie d’un groupe de Lie G on associe un semi-groupe complexe S=G Exp (iW) qui permet une action holomorphe de G×G. Si W est l’algèbre de Lie toute entière, le semi-groupe S est un groupe complexe réductif. Dans cet article on montre que chaque semi-groupe S est une variété de Stein, qu’un domaine biinvariant DS est de Stein si et seulement si D=G Exp (D h )D h iW est convexe, que toute fonction holomorphe sur D s’étend au plus petit domaine de Stein contenant D, et que les fonctions biinvariantes plurisousharmoniques sur D correspondent aux fonctions convexes sur D h .

     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},
     zbl = {0901.22003},
     mrnumber = {99e:22013},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1614/}
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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.1614/

[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. | MR | Zbl

[Fe94] G. Fels, Differentialgeometrische Charakterisierung invarianter Holomorphiegebiete, Schriftenreihe des Graduiertenkollegs “Geometrische und mathematische Physik”, Universität Bochum, 7, 1994. | Zbl

[He93] P. Heinzner, Equivariant holomorphic extensions of real analytic manifolds, Bull. Soc. Math. France, 121 (1993), 445-463. | EuDML | Numdam | MR | Zbl

[HHL89] J. Hilgert, K.H. Hofmann and J.D. Lawson, “Lie Groups, Convex Cones, and Semigroups”, Oxford University Press, 1989. | MR | Zbl

[HiNe93] J. Hilgert and K.-H. Neeb, “Lie semigroups and their applications”, Lecture Notes in Math., 1552, Springer, 1993. | Zbl

[HNP94] J. Hilgert, K.-H. Neeb and W. Plank, Symplectic convexity theorems and coadjoint orbits, Comp. Math., 94 (1994), 129-180. | EuDML | Numdam | MR | Zbl

[Hö73] L. Hörmander, An introduction to complex analysis in several variables, North-Holland, 1973. | Zbl

[Las78] M. Lasalle, 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] Y. Matsushima, and A. Morimoto, “Sur certains espaces fibrés holomorphes sur une variété de Stein”, Bull. Soc. Math. France, 88 (1960), 137-155. | EuDML | Numdam | MR | Zbl

[Ne94a] K.-H. Neeb, Holomorphic representation theory II, Acta Math., 173-1 (1994), 103-133. | MR | Zbl

[Ne94b] K.-H. Neeb, Realization of general unitary highest weight representations, Preprint 1662, Technische Hochschule Darmstadts, 1994.

[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. | Zbl

[Ne94d] K.-H. Neeb, A convexity theorem for semisimple symmetric spaces, Pacific Journal of Math., 162-2 (1994), 305-349. | MR | Zbl

[Ne94e] K.-H. Neeb, On closedness and simple connectedness of adjoint and coadjoint orbits, Manuscripta Math., 82 (1994), 51-65. | MR | Zbl

[Ne95a] K.-H. Neeb, Holomorphic representation theory I, Math., Ann., 301 (1995), 155-181. | MR | Zbl

[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. | MR | Zbl

[Ne96a] K.-H. Neeb, Invariant Convex Sets and Functions in Lie Algebras, Semigroup Forum 53 (1996), 230-261. | MR | Zbl

[Ne96b] K.-H. Neeb, Coherent states, holomorphic extensions, and highest weight representations, Pac. J. Math., 174-2 (1996), 497-542. | Zbl

[Ne98] K.-H. Neeb, “Holomorphy and Convexity in Lie Theory”, de Gruyter, Expositions in Mathematics, to appear. | Zbl

[Ra86] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, New York, 1986. | MR | Zbl

[Ro63] H. Rossi, On Envelopes of Holomorphy, Comm. on Pure and Appl. Math., 16 (1963), 9-17. | MR | Zbl

Cited by Sources: