Convexity properties of gradient maps associated to real reductive representations

Leonardo Biliotti

created by daniele on 27 Sep 2019 modified on 22 Oct 2019

25 oct 2019
-- 14:30

Aula Tricerri, DIMAI, Firenze

Abstract.

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.