Plurisubharmonic functions and the Kempf-Ness theorem.(English)Zbl 0795.32002

This paper completes the investigation of the authors begun in Indiana Univ. Math. J. 39, No. 1, 61-65 (1990; Zbl 0709.32011)]. Its main results are the following:
(A) Let $$G$$ be a complex reductive group, $$K$$ a maximal compact subgroup of $$G$$ and $$H$$ a closed complex subgroup of $$G$$. If $$\varphi$$ is a $$K$$- invariant strictly plurisubharmonic function on $$G/H$$ with a critical point, then $$H$$ is reductive and $$\varphi$$ is proper.
(B) For a closed subgroup $$L$$ (not necessarily connected) of a compact connected group $$K$$, an $$L$$-invariant function on $$K^ \mathbb{C}/L^ \mathbb{C}$$ has $$x_ 0=eL^ \mathbb{C}$$ as a critical point if and only if its restriction to $$N(L^ \mathbb{C})/L^ \mathbb{C}$$ has $$x_ 0$$ as a critical point. In particular if $$N(L^ \mathbb{C})/ L^ \mathbb{C}$$ is finite, then any $$L$$-invariant function on $$K^ \mathbb{C}/L^ \mathbb{C}$$ has a critical point.
These results are inspired by the Kempf-Ness theorem [G. Kempf and L. Ness, Lect. Notes Math. 732, 233-243 (1979; Zbl 0407.22012)] and its various applications. The proof of theorem (A) is based on the natural convexity properties of plurisubharmonic functions and a result of G. D. Mostow [Am. J. Math. 77, 247-278 (1955; Zbl 0067.160)] as opposed to the Hilbert-Mumford theorem and properties of parabolic subgroups of algebraic groups which were the main tools in the original Kempf-Ness theorem. However, the results are stronger and apply to manifolds which have a strictly plurisubharmonic function; in particular they apply to Stein manifolds and give the main results on orbit closures of reductive groups operating on Stein manifolds which were obtained originally by R. W. Richardson [Math. Ann. 208, 323-331 (1974; Zbl 0276.32021)]. Amongst other applications, results of D. Luna [Invent. Math. 29, 231-238 (1975; Zbl 0315.14018)] in the complex analytic category are recovered.

MSC:

 32U05 Plurisubharmonic functions and generalizations 32M10 Homogeneous complex manifolds
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