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Proper Lie group actions on real-analytic manifolds. (Eigentliche Wirkungen von Liegruppen auf reell-analytischen Mannigfaltigkeiten.) (German) Zbl 0840.22017

Schriftenreihe des Graduiertenkollegs Geometrie und Mathematische Physik. Heft 5. Bochum: Math. Inst., Univ. Bochum, 53 S. (1994).
Let \(G\) be a real Lie group and \(X\) a connected analytic \(G\)-manifold. The main objective of this dissertation is to study the existence of “holomorphic extensions” of \(X\) in the following sense. A complex \(G\)-space \(X^*\), i.e., \(G\) acts in an analytic fashion by holomorphic automorphisms, is called a \(G\)-complexification of \(X\) if there exists a real analytic equivariant map \(i : X \to X^*\) such that for each equivariant analytic map \(\varphi\) in some complex \(G\)-space \(Y\) there exists an open invariant neighborhood \(U^*\) of \(i(X)\) in \(X^*\) and a holomorphic equivariant map \(\varphi^* : U^* \to Y\) with \(\varphi^* \circ i = \varphi \), and \(\varphi^*\) is unique in the sense that for each open invariant neighborhood \(V^*\) of \(i (X)\) and each holomorphic equivariant map \(\psi^* : V^* \to Y\) with \(\psi^* \circ i = \varphi\) the mappings \(\varphi^*\) and \(\psi^*\) coincide on an invariant neighborhood of \(i(X)\). If the \(G\)-complexification \(X^*\) has the property that \(i : X \to X^*\) is a diffeomorphism onto a closed submanifold, then \(X^*\) is called a \(G\)-extension. Such extensions do not exist in general, but one of the main results of the thesis under review states that whenever the \(G\)-action on \(X\) is proper, then \(X\) has a \(G\)-extension on which \(G\) acts in a proper way (cf. Section 2.4). If, in addition, \(G\) has only finitely many connected components and \(K\) is a maximal compact subgroup (which exists under this hypothesis), then it is shown that the properness of the action on \(X\) implies the existence of a global slice \(S\) in the analytic category such that \(X \cong G \times_KS\) holds in an analytic equivariant fashion. The main new point in this result is the passage from the differentiable to the analytic category. In the differentiable case it is a theorem of H. Abels [Math. Ann. 212, 1-19 (1974; Zbl 0287.57018)]. This result has already a number of interesting applications such as the existence of an invariant analytic Riemannian metric. In the third part the author generalizes a theorem of H. Grauert [Ann. Math., II. Ser. 68, 460-472 (1958; Zbl 0108.07804)] to the equivariant case where it states that whenever the action of \(G\) on \(X\) is proper, there exists a \(G\)-extension which is Stein such that \(i(X) \subset X^*\) is a strict \(G\)-equivariant deformation retract of \(X^*\).

MSC:

22E30 Analysis on real and complex Lie groups
32D15 Continuation of analytic objects in several complex variables
32M05 Complex Lie groups, group actions on complex spaces
57S20 Noncompact Lie groups of transformations
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