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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 $G \times G$ 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 $G Exp (D_{h})$ , with $D h \subseteq i W$ convex, that each holomorphic function on $D$ extends to the smallest biinvariant Stein domain containing $D$ ,...

On the complex geometry of invariant domains in complexified symmetric spaces

Karl-Hermann Neeb (1999)

Annales de l'institut Fourier

Let $M = G / H$ be a real symmetric space and $𝔤 = 𝔥 + 𝔮$ the corresponding decomposition of the Lie algebra. To each open $H$ -invariant domain $D_{𝔮} \subseteq i 𝔮$ consisting of real ad-diagonalizable elements, we associate a complex manifold $Ξ (D_{𝔮})$ which is a curved analog of a tube domain with base $D_{𝔮}$ , and we have a natural action of $G$ by holomorphic mappings. We show that $Ξ (D_{𝔮})$ is a Stein manifold if and only if $D_{𝔮}$ is convex, that the envelope of holomorphy is schlicht and that $G$ -invariant plurisubharmonic functions correspond to convex $H$ -invariant...

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On automorphism groups of connected Lie groups.

On c-actions on compact complex manifolds with many compact orbits.

On Locally Symmetrie Homogeneous Domains of Completely Reducible Linear Lie Groups.

On observable subgroups of complex analytic groups and algebraic structures on analytic homogeneous spaces.

On Ol'shanskii's semigroup.

On porcupine varieties in Lie algebras.

On the complex and convex geometry of Ol'shanskii semigroups

On the complex geometry of invariant domains in complexified symmetric spaces

On the holomorphic separabilitiy of discrete quotients of complex Lie groups.

On the order of the automorphism group of a surface of general type.

On the set of orbits for a Borel subgroup.

On the Torelli problem of Abelian complex Lie groups.

Currently displaying 1 – 12 of 12