Convexity properties of gradient maps associated to real reductive representations

Let G=Kexp(p) 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 (1984), see also Berline and Vergne (2011). If G is not Abelian, we explicitly compute the image of the gradient map with respect to A=exp(a), where a⊂p is an Abelian subalgebra, of the gradient map restricted on the closure of a G orbit. We also describe the convex hull of the image of the gradient map, with respect to G, restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert–Mumford criterion for real reductive Lie groups stressing the properties of the Kempf–Ness functions and applying the stratification theorem proved in Heinzner et al. (2008).