相关论文: Orbits of real forms in complex flag manifolds

Let $\mathcal{G} \rightrightarrows M$ be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection $M \to M/\mathcal{G}$ is a fibration if and only if $\mathcal{G} \rightrightarrows M$ is…

It is shown that the cycle space of an arbitrary orbit of a non-Hermitian real form G in a flag manifold $Z=G^\mathbb C/Q$ of its complexification is naturally equivalent to a certain universal domain which depends only on G. This makes use…

We deal with compact Kaehler manifolds M which are acted on by a semisimple compact Lie group G of isometries with codimension one regular orbits. We provide an explicit description of the standard blow-ups of such manifolds along complex…

We compute the Euler-Poincar\'e characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

Real flag manifolds are the isotropy orbits of noncompact symmetric spaces $G/K$. Any such manifold $M$ enjoys two very peculiar geometric properties: It carries a transitive action of the (noncompact) Lie group $G$, and it is embedded in…

We classify all compact simply connected homogeneous CR manifolds $M$ of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal…

Let $H$ be a connected spherical subgroup of a semisimple algebraic group $G$. In this paper, we give a criterion for $H$-orbit closures in the flag variety of $G$ to have nice geometric and cohomological properties. Our main tool is the…

We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex…

Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $\Gamma\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/\Gamma\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $\Gamma$ in…

We consider aspects of the relationship between nilpotent orbits in a semisimple real Lie algebra $\mathfrak{g}$ and those in its complexification $\mathfrak{g}_{\mathbb{C}}$. In particular, we prove that two distinct real nilpotent orbits…

Complex geometric properties of the orbits of a non-compact real form $G_0$ in a flag manifold $Z=G/Q$ of a complex semi-simple groups $G=G_0^\mathbb C$ are studied. Schubert varieties are used to construct a complex submanifold with…

We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold.…

Consider a simple complex Lie group $G$ acting diagonally on a triple flag variety $G/P_1\times G/P_2\times G/P_3$, where $P_i$ is parabolic subgroup of $G$. We provide an algorithm for systematically checking when this action has finitely…

Consider a smooth manifold and an action on it of a compact connected Lie group with a bi-invariant metric. Then, any orbit is an embedded submanifold that is isometric to a normal homogeneous space for the group. In this paper, we…

We examine the orbits of the (complex) symplectic group, $Sp_n$, on the flag manifold, $\mathscr{F}\ell(\mathbb{C}^{2n})$, in a very concrete way. We use two approaches: we Gr\"obner degenerate the orbits to unions of Schubert varieties…

Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a…

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics…

We study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic, we show that there is a geometrically defined bijection…

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were…

A flag manifold over a semifield K can be partitioned into "half i-circles" which are orbits of a K-action on that flag manifold. Here i is fixed and it corresponds to a simple reflection in the Weyl group. We prove (for certain K) a…