On the complex and convex geometry of Ol'shanskii semigroups

Neeb, Karl-Hermann

Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 149-203.

À tout cône ouvert elliptique convexe $W$ dans l’algèbre de Lie d’un groupe de Lie $G$ on associe un semi-groupe complexe $S=G\mathrm{Exp}\left(iW\right)$ qui permet une action holomorphe de $G\times G$. 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 $D\subseteq S$ est de Stein si et seulement si $D=G\mathrm{Exp}\left(D_h\right)$ où $D_h\subseteq iW$ 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 $D_h$.

To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G\mathrm{Exp}\left(iW\right)$ 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 $G\mathrm{Exp}\left(D_h\right)$, 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 $D_h$.

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