25 oct 2019 -- 14:30
Aula Tricerri, DIMAI, Firenze
Let G=K\exp(\liep) 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. If G is not Abelian, we explicitly compute the image of the gradient map with respect to A=\exp(\lia), where \lia \subset \liep 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. Finally we investigate the natural G action on the projective space P(V) and we discuss some new results which are still in working in progress.