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

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {1},
     year = {1998},
     pages = {149-203},
     doi = {10.5802/aif.1614},
     zbl = {0901.22003},
     mrnumber = {99e:22013},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_1998__48_1_149_0/

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