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Minimum vectors for real reductive algebraic groups. (English) Zbl 0675.14020
In this article, orbits of real reductive groups G acting on real affine varieties X are investigated. The main result is a generalization of previous results of Kempf-Ness in the complex case [cf. G. Kempf and L. Ness in Algebraic geometry, Proc. Summer. Meet., Copenhagen 1978, Lect. Notes Math. 732, 233-243 (1979; Zbl 0407.22012)]. For example, it is proved that the points of a closed G-orbit with minimal distance to zero in a real linear representation X form an orbit under a maximal compact subgroup K of G.
Several applications are given, among them a new proof of the G- properness of the algebraic quotient $$X\to X/G$$ in the real Hausdorff- topology (i.e. the image of a closed G-stable subset of X is closed in X/G) and a reduction of G-conjugacy of reductive subgroups of G to K- conjugacy.
Reviewer: P.Slodowy

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 14P99 Real algebraic and real-analytic geometry 14M15 Grassmannians, Schubert varieties, flag manifolds
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