Satake-Furstenberg compactifications and gradient map

Abstract

Let G be a real semisimple Lie group with finite center and let g = k⊕p be a Cartan decomposition of its Lie algebra. Let K be a maximal compact subgroup of G with Lie algebra k and let τ be an irreducible representation of G on a complex vector space V . We denote by µp : P(V ) −→ p the G-gradient map and by O the unique closed orbit of G in P(V ) which is a K-orbit [33,40]. 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 E = conv(µp(O)). Moreover, any parabolic subgroup of G admits a unique closed orbit which is well-adapted to O and µp respectively. These results are new also in the complex reductive case. The connection between 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 O, we construct a map Ψγ from the Satake compactification of G/K associated to τ and E . If γ is a K-invariant measure then Ψγ is an homeomorphism of the Satake compactification and E . Finally, we prove that for a large class of measures the map Ψγ is surjective.

Compact orbits of Parabolic subgroups

“…The following result is probably well-known. A proof is given in [5]. For sake of completeness we give a proof.…”

Projective representations of real reductive Lie groups and the gradient map

“…Therefore (V, J) is a complex vector space and the G-action on V preserves J. This case has been extensively studied in [10]. We may also assume, up to conjugate, that V = R n and •, • is the canonical scalar product.…”