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On the complex and convex geometry of Ol'shanskii semigroups

Karl-Hermann Neeb (1998)

Annales de l'institut Fourier

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 × 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 ⊆ S is Stein is and only if it is of the form G Exp ( D h ) , with D h ⊆ 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 ⊆ 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...

On the CR-structure of certain linear group orbits in infinite dimensions

Wilhelm Kaup (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits  M explicitly and show as main result that every continuous CR-function on  M has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite...

On the existence of parabolic actions in convex domains of ℂ k + 1

François Berteloot, Ninh Van Thu (2015)

Czechoslovak Mathematical Journal

We prove that the one-parameter group of holomorphic automorphisms induced on a strictly geometrically bounded domain by a biholomorphism with a model domain is parabolic. This result is related to the Greene-Krantz conjecture and more generally to the classification of domains having a non compact automorphisms group. The proof relies on elementary estimates on the Kobayashi pseudo-metric.

On the Faraut-Koranyi hypergeometric functions in rank two

Miroslav Engliš, Genkai Zhang (2004)

Annales de l’institut Fourier

We give a complete description of the boundary behaviour of the generalized hypergeometric functions, introduced by Faraut and Koranyi, on Cartan domains of rank 2. The main tool is a new integral representation for some spherical polynomials, which may be of independent interest.

On weighted Bergman kernels of bounded domains

Sorin Dragomir (1994)

Studia Mathematica

We build on work by Z. Pasternak-Winiarski [PW2], and study a-Bergman kernels of bounded domains Ω ⊂ ℂ N for admissible weights a ∈ L ¹ ( Ω ) .

Optimal destabilizing vectors in some Gauge theoretical moduli problems

Laurent Bruasse (2006)

Annales de l’institut Fourier

We prove that the well-known Harder-Narsimhan filtration theory for bundles over a complex curve and the theory of optimal destabilizing 1 -parameter subgroups are the same thing when considered in the gauge theoretical framework.Indeed, the classical concepts of the GIT theory are still effective in this context and the Harder-Narasimhan filtration can be viewed as a limit object for the action of the gauge group, in the direction of an optimal destabilizing vector. This vector appears as an extremal...

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