2020/12/29 by Biliotti, Leonardo
#14L24 #53D20 #Differential Geometry (math.DG) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2012.14858
Let G be a real semisimple Lie group with finite center and let \mathfrak g=\mathfrak k ⊕ \mathfrak p be a Cartan decomposition of its Lie algebra. Let K be a maximal compact subgroup of G with Lie algebra \mathfrak k and let τ be an irreducible representation of G on a complex vector space V. Let h be a Hermitian scalar product on V such that τ(G) is compatible with respect to U(V,h)\mathbb C. We denote by μ\mathfrak p:\mathbb P(V) \longrightarrow \mathfrak p the G-gradient map and by \mathcal O the unique closed orbit of G in \mathbb P(V), which is a K-orbit, contained in the unique closed orbit of the Zariski closure of τ(G) in U(V,h)\mathbb C. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of G induced by τ are completely determined by the facial structure of the polar orbitope \mathcal E=conv(μ\mathfrak p (\mathcal O)). Moreover, any parabolic subgroup of G admits a unique closed orbit which is well-adapted to \mathcal O and μ\mathfrak p respectively. These results are new also in the complex reductive case. The connection between \mathcal E and τ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure γ on \mathcal O, we construct a map Ψγ from the Satake compactification of G/K associated to τ and \mathcal E. If γ is a K-invariant measure then Ψγ is an homeomorphism of the Satake compactification and \mathcal E. Finally, we prove that for a large class of measures the map Ψγ is surjective.