##
**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 \(X^*\).

Reviewer: K.-H.Neeb (Erlangen)

### 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 |