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. 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 resul

Biliotti, Leonardo

English

arXiv

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)

Convexity properties of gradient maps associated to real reductive representations

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