Spherical gradient manifolds

Abstract

18 pagesNous étudions l'action d'un groupe réel-réductif $G=K\exp(\lie{p})$ sur une sous-variété réel-analytique $X$ d'une variété kählérienne $Z$. Nous supposons que l'action de $G$ peut être prolongée à une action holomorphe du groupe complexifié $G^\mbb{C}$ telle que l'action d'un sous-groupe maximal compact de $G^\mbb{C}$ soit hamiltonienne. L'application moment $\mu$ induit une application gradient $\mu_\lie{p}\colon X\to\lie{p}$. Nous montrons que $\mu_\lie{p}$ separe les orbites de $K$ si et seulement si un sous-groupe minimal parabolique de $G$ possède une orbite ouverte dans $X$. Ce résultat généralise la charactérisation de Brion des variétés kählériennes sphériques qui admettent une application moment

Classification of spherical algebraic subalgebras of real simple Lie algebras of rank 1

“…Let H ⊂ G be a closed subgroup such that X = G/H is a G-gradient manifold with gradient map µ p : X → p. We say that X = G/H is spherical if a minimal parabolic subgroup of G has an open orbit in X. In this case we call H a spherical subgroup of G and h a spherical subalgebra of g. As shown in [MS10] sphericity of X is equivalent to the fact that µ p almost separates the K-orbits in X, i.e., that the map X/K → p/K induced by µ p : X → p has discrete fibers.…”

“…In [HS07] it has been shown that the so-called gradient maps are the right analogue for moment maps when one is interested in actions of a real reductive group G = K exp(p). Spherical gradient manifolds have been introduced in [MS10] in order to carry over Brion's theorem to the real reductive case. To be more precise, we call a G-gradient manifold X ⊂ Z with gradient map µ p : X → p spherical if a minimal parabolic subgroup of G has an open orbit in X.…”

“…If G is connected complex reductive, then a minimal parabolic subgroup is the same as a Borel subgroup of G, so that there is no ambiguity in this definition. The main result of [MS10] states that X is spherical if and only if µ p almost separates the K-orbits in X.…”

“…As the main result of this paper we describe the non-reductive spherical algebraic subalgebras of g where g is a simple Lie algebra of real rank 1 by methods in the spirit of [MS10]. We then apply this result to classify the reductive spherical subalgebras of g, thus obtaining a second proof of the rank one case in [KKPS16].…”

Moment polytopes in real symplectic geometry I

“…Since the founding article of R. Richardson and P. J. Slodowy [20] where they showed that the Kempf-Ness theorem extends to representations of real reductive groups, many authors have studied extensions of geometric invariant theory to the real framework [6,8,15,4,3].…”