# Questions tagged [homogeneous-spaces]

The homogeneous-spaces tag has no usage guidance.

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### Conformally flat homogeneous spaces

Let's say we have a homogeneous space $H\backslash G$.
Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?
I am particularly ...

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**1**answer

168 views

### Compact simply-connected homogeneous symplectic manifold

I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...

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### The isometry groups of flag manifolds

For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ ...

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68 views

### Classification of Euclidian-like Klein geometries in spirit of Erlangen program

All we know about Erlangen Program approach to geometry. The main idea is to consider Lie Group $G$ acting transitively on smooth manifold $M$ (e.g. nLab). I really like such style of thinking, but ...

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385 views

### Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice.
I have heard that, for any real number $\...

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629 views

### Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?

Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...

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**1**answer

90 views

### Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$

Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram
$$
\beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,.
$$
Let $P_{\{\beta_2\}}...

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51 views

### Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?

$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...

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58 views

### Construction of homogeneous space

Given a Hilbertspace $\mathcal{H}$ of dimension $n<\infty$ and all hermitian matricies $Symm(\mathcal{H})$. I'd guess that the set $M_{2,2} \subset Symm(\mathcal{H})$ of all matricies of rank 4 and ...

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49 views

### Injectivity of exponential chart in a homogeneous space

Let $G$ be a finite-dimensional connected real Lie group and $\mathfrak{g}$ its Lie algebra. Let $\exp : \mathfrak{g} \to G$ denote the exponential map. Among the results proved independently by ...

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280 views

### Explicit description of SU(2,2)/U

Consider the real diagonal $4\times 4$ - matrix
$$I_{2,2}={\rm diag}(1,1,-1,-1)$$
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\...

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99 views

### Integrals of modular forms

Setup: Write $G = \text{SL}_2(\mathbf{R})$ and $\Gamma = \text{SL}_2(\mathbf{Z})$.
Let $f$ be a modular form on $\mathbf{H}$ of weight $2k$, so that
$$f(gz) = f(z) (cz + d)^{2k} \qquad \text{for} \...

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55 views

### Criterion for existence of a homogeneous space associated to a Lie pair

Recall that every finite-dimensional real Lie algebra $\mathfrak{g}$ is the Lie algebra of a simply-connected Lie group, which is unique up to isomorphism.
This statement generalises somewhat to ...

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127 views

### Is there a class of non-symmetric spaces $G/SO(n)$ such that $Hol(G/SO(n))=SO(n)$?

Is there a certain class (or unique construction of a set) of simply connected non-symmetric spaces $G/SO(n)$ such that the Riemannian holonomy is $Hol(G/SO(n))=SO(n)$? (That is, is there a way to ...

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133 views

### Torus actions on $Sp(n)$-spheres

In this old question of mine
https://math.stackexchange.com/questions/1651906/spheres-as-symplectic-homogeneous-spaces
the presentation of spheres as symplectic group homogeneous spaces was ...

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132 views

### Can we see the symmetry of the quantum Schubert polynomial of a point

Let $X=G/B$ be a homogeneous space and consider the quantization map
$$
S_W\otimes\mathbb{C}[q]\to(S(\mathfrak{h})\otimes\mathbb{C}[q])/I_W^q\,,
$$
where
$S_W$ is the coinvariant algebra of the Weyl ...

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102 views

### A section over an orbit space

Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...

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**1**answer

243 views

### Gelfand pairs and (self)-dual representations

For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of ...

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180 views

### Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$

The ideal of the $10$-dimensional Spinor variety $S^{10}\subset\mathbb{P}^{15}$ is generated by $10$ quadrics.
Does anyone know a reference where these 10 quadratic equations are written down ...

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60 views

### Submanifolds of nilmanifolds coming from Lie subgroups

Let $G$ be a connected simply connected nilpotent real Lie group and $\Gamma$ a lattice in $G$, such that $M=\Gamma \backslash G$ is a compact nilmanifold. Let $p:G \to M$ be the projection. If $S$ is ...

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54 views

### evolution of Grassmannians along geodesic line

Let $p_0$, $p_1$ be two $n \times 2$ orthonormal matrices that represent two points on the real $Gr_{2,n}$, i.e. two 2-d subspaces in $\mathbb{R}^n$. Let $p(t): [0,1] \rightarrow Gr_{2,n}$ be a ...

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95 views

### Toral subgroup acting regularly on the homogeneous space

Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...

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78 views

### Compactification of symmetric spaces

Let $G = GL_n$. In the literature, I see that the symmetric space associated to $G$ is of the form $G(\mathbb{R})/K_\infty$ with $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. ...

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140 views

### Degeneration of coadjoint orbits

Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits ...

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294 views

### Is Bing's countable connected space topologically homogeneous?

In this paper R.H. Bing has constructed his famous example of a countable connected Hausdorff space.
The Bing space $\mathbb B$ is the rational half-plane $\{(x,y)\in\mathbb Q\times \mathbb Q:y\ge 0\...

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84 views

### Local vs global homogeneity of topological spaces

I am interested in the relation between global and local homogeneity of topological spaces. On one extreme we have rigid spaces, i.e., topological spaces with trivial homeomorphism group.
Question. ...

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439 views

### The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$

The branching laws for the restricted representation of $SO(n)$ with respect to the subgroup $SO(n-1)$ are discussed in this Wikipedia article. Am I correct in reading from this that any given ...

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125 views

### A criterion for a $G$-variety to be isomorphic to $G/H$

Let $k$ be an algebraically closed field of characteristic 0.
Let $G$ be a connected linear algebraic group over $k$.
Let $H\subset G$ be an algebraic $k$-subgroup.
Let $Y$ be an algebraic variety ...

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64 views

### Decomposition of conic equation for two intersecting lines [closed]

By using homogeneous coordinates ${\bf x}=[x, y, 1]^T$, a conic section can be expressed by ${\bf x}^T {\bf C} {\bf x} = 0$ with ${\bf C}$ being a 3x3 symmetric matrix. A degenerate form of the conic ...

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59 views

### dense orbit projective dual homogeneous space

Let $\mathrm{G}$ be a semi-simple algebraic group over $\mathbb{C}$ and $V$ be a finite dimensional representation of $\mathrm{G}$. Let $x \in V$ be a non zero vector such that the variety $\mathrm{G}....

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### Criterion for homogeneity

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and let $G=Aut(\Omega)$ be the full group of self-biholomorphisms of $\Omega$. Assume that there is $z\in \Omega$, such that the orbit of $z$ is ...

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74 views

### Criterion of holomorphy

I have posted a similar question on MSE few days ago, but it received no attention.
Let $G$ be a homogeneous (or even symmetric) domain in $\mathbb{C}^{n}$ and $u:G\to \mathbb{C}\backslash\{0\}$.
Q1....

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### Holomorphic extension of an action by a compact Lie group on a complex homogeneous manifold

Let $G$ be a compact Lie group and let $M$ be a $G$-homogeneous manifold. Suppose that $M$ is endowed with a complex structure invariant by the action of $G$. Denote by $G_{\mathbb C}$ the ...

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254 views

### Compact Lie group inclusions that are trivial on all homotopy groups

Is there a classification of closed subgroups $H\le SO(n)$ such that the inclusion $H\to SO(n)$ is trivial on all homotopy groups?
This happen e.g. when the group $H$ is finite. Are there other ...

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134 views

### The group of automorphisms a pair $(G,X)$ where $X$ is a spherical homogeneous space of $G$

I wish to "compute" ${{\rm Aut}}(G,X)$ for a spherical homogeneous space $X$ of $G$ in terms of the spherical datum of $X$.
First let $G$ be any algebraic group over $\mathbb C$, and let $X$ be any ...

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158 views

### Homogenous structure on $S^2\times S^2$ and its geometry

Is a well known fact that $SO(3)$ acts transitively on $S^2$ and that the isotropy group of this action is $SO(2).$ In this case, $S^2$ has a natural structure of homogeneous space. In particular, I ...

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87 views

### Isotropy symmetric holomorphic functions

Let $G$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $z\in G$.
Assume that $f$ is a holomorphic function on $G$, which is isotropy symmetric, i.e. $f\circ \varphi=f$ for any ...

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317 views

### Explicit description of the Lagrangian Grassmannian as a homogeneous space

Looking at this and this question about the Lagrangian Grassmannian, and its linked Wikipedia description as the quotient of $Sp(N)$ by the unitary group $U(n)$, I wondering what is the explicit ...

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139 views

### Flag manifolds for classical groups

Let $G$ be a compact connected Lie group and $T$ be a maximal torus in $G$. Then the homogeneous space $G/T$ is a simply connected orientable manifold. (See, e.g., Hofmann-Morris: The structure of ...

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290 views

### Combinatorics of the Cohomology Ring of the Lagrangian Grassmannians

The Lagrangian Grassmannian is an important example in symplectic geometry, see here or here for details. It shares many similarities with the ordinary Grassmannians (as one would expect from the name)...

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### Gradient of spectral function on noncompact homogeneous space

Let $(M,g)$ be a noncompact Riemannian manifold whose isometry group acts transitively on $M$, i.e. a (not necessarily normal) homogeneous space. Let $e_{\lambda}(x,y)$ be the integral kernel of
$f \...

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104 views

### When de Rham cohomology classes of real Grassmanian manifold are given by algebraic expressions?

Let $G(m,n)$ be a real oriented Grassmanian of oriented $m$ planes in $\mathbb{R}^{n+m}$. Real de Rham cohomology classes of this space can be represented by $SO(n+m)-$invariant differential forms. ...

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207 views

### example of an n-transitive but not infinitely transitive group action on a space

Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ ...

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394 views

### Applications of Schubert calculus

Schubert calculus is a venerable field in mathematics where the object of study is the cohomology ring of the Grassmannians. Since it has been around for over a hundred years one might wonder if any ...

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279 views

### Homogeneous spaces which are not torus bundle over flag manifolds

For a compact semisimple Lie group $G$, what is an example of a homogeneous space of $G$ which is not a torus bundle over a generalized flag manifold of $G$. Examples for $SU(N)$ would be of most ...

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103 views

### Dimension of the orbit of a curve in the moduli space of stable maps

Let $X=G/P$ be a homogeneous space where $G$ is a connected, simply connected, simple (in the sense that the root system $R$ is irreducible), complex, linear algebraic group and $P$ is a parabolic ...

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86 views

### Two questions on homogeneous domains

Let $G$ be a domain in $\mathbb{C}^{n}$ and let $Aut(G)$ be the group of biholomorphic selfmaps of $G$. $G$ is called:
(1) homogeneous if $Aut(G)$ acts transitively on $G$, i.e. for any $z,w\in G$ ...

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262 views

### $K$-Theory Of Aloff--Wallach Spaces

Aloff--Wallach spaces were discussed in this question. They are quotients of SU(3) by U(1) indexed by a lattice of rank 2. Am I correct in guessing that the $K$-theory group $K_0$ of these spaces is ...

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710 views

### Wonderful compactification

Suppose $G$ is a semi-simple group of adjoint type over an algebraic closed field, and $X$ its wonderful compactification a la De Concini and Procesi. Let $P=MU$ be a parabolic subgroup in $G$, and ...

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160 views

### k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$.
Question. Are ...