Spherical Stein manifolds and the Weyl involution

Dmitri Akhiezer[1]

  • [1] Institute for Information Transmission Problems B. Karetny per. 19 Moscow, 127994 (Russia)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 3, page 1029-1041
  • ISSN: 0373-0956

Abstract

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We consider an action of a connected compact Lie group on a Stein manifold by holomorphic transformations. We prove that the manifold is spherical if and only if there exists an antiholomorphic involution preserving each orbit. Moreover, for a spherical Stein manifold, we construct an antiholomorphic involution, which is equivariant with respect to the Weyl involution of the acting group, and show that this involution stabilizes each orbit. The construction uses some properties of spherical subgroups invariant under certain real automorphisms of complex reductive groups.

How to cite

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Akhiezer, Dmitri. "Spherical Stein manifolds and the Weyl involution." Annales de l’institut Fourier 59.3 (2009): 1029-1041. <http://eudml.org/doc/10415>.

@article{Akhiezer2009,
abstract = {We consider an action of a connected compact Lie group on a Stein manifold by holomorphic transformations. We prove that the manifold is spherical if and only if there exists an antiholomorphic involution preserving each orbit. Moreover, for a spherical Stein manifold, we construct an antiholomorphic involution, which is equivariant with respect to the Weyl involution of the acting group, and show that this involution stabilizes each orbit. The construction uses some properties of spherical subgroups invariant under certain real automorphisms of complex reductive groups.},
affiliation = {Institute for Information Transmission Problems B. Karetny per. 19 Moscow, 127994 (Russia)},
author = {Akhiezer, Dmitri},
journal = {Annales de l’institut Fourier},
keywords = {Reductive groups; spherical subgroups; spherical Stein manifolds; antiholomorphic involutions; reductive groups},
language = {eng},
number = {3},
pages = {1029-1041},
publisher = {Association des Annales de l’institut Fourier},
title = {Spherical Stein manifolds and the Weyl involution},
url = {http://eudml.org/doc/10415},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Akhiezer, Dmitri
TI - Spherical Stein manifolds and the Weyl involution
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 3
SP - 1029
EP - 1041
AB - We consider an action of a connected compact Lie group on a Stein manifold by holomorphic transformations. We prove that the manifold is spherical if and only if there exists an antiholomorphic involution preserving each orbit. Moreover, for a spherical Stein manifold, we construct an antiholomorphic involution, which is equivariant with respect to the Weyl involution of the acting group, and show that this involution stabilizes each orbit. The construction uses some properties of spherical subgroups invariant under certain real automorphisms of complex reductive groups.
LA - eng
KW - Reductive groups; spherical subgroups; spherical Stein manifolds; antiholomorphic involutions; reductive groups
UR - http://eudml.org/doc/10415
ER -

References

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  9. Deane Montgomery, Simply connected homogeneous spaces, Proc. Amer. Math. Soc. 1 (1950), 467-469 Zbl0041.36309MR37311
  10. G. D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1955), 44-55 Zbl0065.01404MR69830
  11. Frank J. Servedio, Prehomogeneous vector spaces and varieties, Trans. Amer. Math. Soc. 176 (1973), 421-444 Zbl0266.20043MR320173
  12. É. A. Vinberg, B. N. Kimelʼfelʼd, Homogeneous domains on flag manifolds and spherical subsets of semisimple Lie groups, Funktsional. Anal. i Prilozhen. 12 (1978), 12-19, 96 Zbl0439.53055MR509380

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