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 / Biliotti, L.. - In: JOURNAL OF GEOMETRY AND PHYSICS. - ISSN 0393-0440. - 151(2020), pp. 1-15. [10.1016/j.geomphys.2020.103621]

Convexity properties of gradient maps associated to real reductive representations

Biliotti L.
2020

Abstract

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 / Biliotti, L.. - In: JOURNAL OF GEOMETRY AND PHYSICS. - ISSN 0393-0440. - 151(2020), pp. 1-15. [10.1016/j.geomphys.2020.103621]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11381/2873383
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