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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

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