Truncated Lie groups and almost Klein models (Q867354)

Let \(G\) be a Lie group acting analytically on a manifold \(M\) with non-empty fixed point set. The authors pose the problem of finding a transitive action of a Lie group \(H\) on \(M\) (a `Klein model') such that \(G\) is its isotropy group. The approach of the authors is to associate with a sufficiently nice action \((G,M)\) a so-called almost Klein model, consisting of a finite-dimensional vector space \(\mathcal E\) of complete vector fields on \(M\). The main theorem gives a criterion for the existence of a Klein model, assuming the existence of a semisimple Lie group \(S\) of automorphisms of \((G,M,{\mathcal E})\) with certain properties; namely, the fixed point set of \(S\) is non-empty and discrete, the action of \(S\) on the tangent space of a fixed point fixes no vector other than 0, and the `isotopic dimensions' of \(T_xM\) as an \(S\)-module are greater than 3. The technique employed is that of the embedding problem for truncated Lie algebras, as studied by \textit{V. Guillemin} and \textit{S. Sternberg} [Bull. Am. Math. Soc. 70, 16--47 (1964; Zbl 0121.38801)], \textit{I. Hayashi} [J. Math. Soc. Japan 22, 1--14 (1970; Zbl 0182.36402)], and \textit{C. Fredfield} [Bull. Am. Math. Soc. 76, 331--333 (1970; Zbl 0194.05505)].