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Some transitive linear actions of real simple Lie groups. (English) Zbl 1241.22017

The paper under review deals with transitive actions of real simple Lie groups. It is known that \(\mathrm{Sp}(n,\mathbb R)\) acts transitively on \(\mathbb R^{2n} \setminus \{0\}\) and \(\mathrm{SL}(n,\mathbb R)\) acts transitively on \(\mathbb R^n \setminus \{0\}\). A natural question is whether there are proper connected Lie subgroups of the above groups whose actions are transitive and if yes, which are the minimal ones. The present paper answers this question completely.
Theorem: When \(n\) is odd, no connected Lie subgroup of \(\mathrm{SL}(n, \mathbb R)\) can act transitively on \(\mathbb R^n \setminus \{0\}.\) When \(n = 2k,\) with \(k\) odd, both \(\mathrm{Sp}(k,\mathbb R)\) and \(\mathrm{SL}(k,\mathbb C)\) act transitively on \(\mathbb R^{2k} \setminus \{0\}\) and they are the minimal ones. When \(k = 2m\) and \(n = 4m,\) both \(\mathrm{Sp}( 2m, \mathbb R) \subseteq \mathrm{SL}(4m, \mathbb R)\) and \(\mathrm{SL}(m, \mathbb H) \subseteq \mathrm{SL}(4m, \mathbb R)\) act transitively on \(\mathbb R^{4m} \setminus \{0\}\) and they are the minimal ones.
Main ingredients in the proof are the classification of effective transitive actions of connected compact Lie groups on spheres by Montgomery-Samelson and Borel and Theorem 2.2 in the present paper which states:
Theorem. Suppose \(G\) is a connected Lie subgroup of \(\mathrm{SL}(n,\mathbb R)\) which acts transitively on \(\mathbb R^n \setminus \{0\}\) and is minimal with respect to this property. Then \(G\) is a noncompact simple Lie group.

MSC:

22E46 Semisimple Lie groups and their representations
22F30 Homogeneous spaces
54H15 Transformation groups and semigroups (topological aspects)
57S15 Compact Lie groups of differentiable transformations
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