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Totally real orbits in affine quotients of reductive groups

Published online by Cambridge University Press:  22 January 2016

H. Azad ,
J. J. Loeb  and
M. N. Qureshi
H. Azad
Affiliation:
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
J. J. Loeb
Affiliation:
Université D’Angers, Faculté des Sciences, Department de Mathématiques, 49045 Angers, France
M. N. Qureshi
Affiliation:
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
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Let K be a compact connected Lie group and L a closed subgroup of K In [8] M. Lassalle proves that if K is semisimple and L is a symmetric subgroup of K then the holomorphy hull of any K-invariant domain in Kc/Lc contains K/L. In [1] there is a similar result if L contains a maximal torus of K. The main group theoretic ingredient there was the characterization of K/L as the unique totally real K-orbit in Kc/Lc. On the other hand, Patrizio and Wong construct in [9] special exhaustion functions on the complexification of symmetric spaces K/L of rank 1 and find that the minimum value of their exhaustions is always achieved on K/L. By a lemma of Harvey and Wells [6] one knows that the set where a strictly plurisubharmonic (briefly s.p.s.h) function achieves its minimum is totally real. There is a related result in [2, Lemma 1.3] which states that if φ is any differentiable function on a complex manifold M then the form ddcφ vanishes identically on any real submanifold N contained in the critical set of φ; in particular if φ is s.p.s.h then N must be totally real.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 139 , September 1995 , pp. 87 - 92
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1995

References

[ 1 ] Azad, H. and Loeb, J. J., On a theorem of Kempf and Ness, Indiana University Math. J., 39, No. 1 (1990), 61–65.Google Scholar
[ 2 ] Azad, H. and Loeb, J. J., Plurisubharmonic functions and the Kempf-Ness theorem, Bull. London Math. Soc, 25 (1993), 162–168.Google Scholar
[ 3 ] Bando, S. and Kobayashi, R., Ricci flat Kähler metrics on affine algebraic manifolds. II, Math. Ann., 287 (1990), 175–180.Google Scholar
[ 4 ] Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces, Amer. J. Math., 80 (1958), 459–538.Google Scholar
>[ 5 ] Eliashberg, Y. and Gromov, M., Convex symplectic manifolds, Proc. Symp. Pure Math., Vol. 52, Part 2, 135–162, AMS, 1991.Google Scholar
[ 6 ] Harvey, F. R. and Wells, R. O., Zero sets of non-negative strictly plurisubharmonic functions, Math. Ann., 201 (1973), 165–170.Google Scholar
[ 7 ] Helgason, G., Differential Geometry and Symmetric Spaces, New York, Academic Press, 1978.Google Scholar
[ 8 ] Lassalle, M., Série de Laurent des fonction holomorphes dans la complexification d’un espace symétrique compact, Ann. Scient. Ec. Norm Sup., t. 11 (1978), 167–210.Google Scholar
[ 9 ] Patrizio, G. and Wong, P. M., Stein manifolds with compact symmetric center, Math. Ann., 289 (1991), 355–382.Google Scholar