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**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.

(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.

Reviewer: H.Azad (Islamabad) and J.J.Loeb (Angers)