On the complex and convex geometry of Ol'shanskii semigroups

Karl-Hermann Neeb

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 1, page 149-203
  • ISSN: 0373-0956

Abstract

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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 × 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 , and that biinvariant plurisubharmonic functions on D correspond to invariant convex functions on D h .

How to cite

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Neeb, Karl-Hermann. "On the complex and convex geometry of Ol'shanskii semigroups." Annales de l'institut Fourier 48.1 (1998): 149-203. <http://eudml.org/doc/75274>.

@article{Neeb1998,
abstract = {To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G\{\rm Exp\}(iW)$ 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 $G\{\rm Exp\}(D_h)$, 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 $D_h$.},
author = {Neeb, Karl-Hermann},
journal = {Annales de l'institut Fourier},
keywords = {Ol'shanskii semigroup; classical Lie group; real Lie group; complex reductive Lie group; Lie algebras},
language = {eng},
number = {1},
pages = {149-203},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the complex and convex geometry of Ol'shanskii semigroups},
url = {http://eudml.org/doc/75274},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Neeb, Karl-Hermann
TI - On the complex and convex geometry of Ol'shanskii semigroups
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 1
SP - 149
EP - 203
AB - To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G{\rm Exp}(iW)$ 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 $G{\rm Exp}(D_h)$, 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 $D_h$.
LA - eng
KW - Ol'shanskii semigroup; classical Lie group; real Lie group; complex reductive Lie group; Lie algebras
UR - http://eudml.org/doc/75274
ER -

References

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