Convexity properties of gradient maps associated to real reductive representations

Abstract

Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson [38]. A similar result is proved for the gradient map associated to the natural G action on P(V ). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic result.2010 Mathematics Subject Classification. 22E45,53D20; 14L24.

The Kempf–Ness Theorem and invariant theory for real reductive representations

“…In a recent paper [7], the author gives a proof of the following result avoiding any algebraic result. Applying Theorem 26 and the Slice Theorem given in [21], see also [30,41], we are able to proof the Hilbert-Mumford criterion for reductive groups.…”

“…We point out that in [22] the authors prove that N is real algebraic. If G is Abelian a proof avoiding any algebraic result is given in [7]. However, if G is real reductive, then we are not able to give a new proof of this result.…”

“…We also prove that v minimize the distance to 0 ∈ V within the orbit G • x (Corollary 23). Using results proved in [7] and the Slice Theorem we prove the Hilbert Mumford criterion for real reductive representations (Theorems 27 and 28). Finally we prove that the null cone is closed (Theorem 29).…”

Moment polytopes in real symplectic geometry I

“…Since the founding article of R. Richardson and P. J. Slodowy [20] where they showed that the Kempf-Ness theorem extends to representations of real reductive groups, many authors have studied extensions of geometric invariant theory to the real framework [6,8,15,4,3].…”