Convexity properties of gradient maps associated to real reductive representations
| Autor: | Leonardo Biliotti |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Convex hull
Pure mathematics 010102 general mathematics Subalgebra Closure (topology) 22E45 53D20 14L24 General Physics and Astronomy Lie group Function (mathematics) 01 natural sciences Convexity 0103 physical sciences FOS: Mathematics 010307 mathematical physics Geometry and Topology 0101 mathematics Abelian group Representation Theory (math.RT) Mathematics - Representation Theory Mathematical Physics Vector space Mathematics |
| DOI: | 10.48550/arxiv.1905.01915 |
| Popis: | Let G = K exp ( 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). |
| Databáze: | OpenAIRE |
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