# Tagged Questions

The homogeneous-spaces tag has no usage guidance.

**11**

votes

**1**answer

278 views

+50

### 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 $\...

**12**

votes

**3**answers

605 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 ...

**3**

votes

**1**answer

86 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\}}...

**2**

votes

**0**answers

50 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{...

**1**

vote

**0**answers

57 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 ...

**5**

votes

**0**answers

40 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 ...

**7**

votes

**2**answers

274 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}\...

**5**

votes

**0**answers

91 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} \...

**7**

votes

**0**answers

54 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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**1**answer

129 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 ...

**5**

votes

**0**answers

129 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 ...

**1**

vote

**0**answers

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:
...

**2**

votes

**1**answer

236 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 ...

**5**

votes

**1**answer

177 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 ...

**2**

votes

**0**answers

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 ...

**2**

votes

**0**answers

53 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 ...

**1**

vote

**0**answers

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 $...

**3**

votes

**0**answers

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 $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. ...

**3**

votes

**1**answer

138 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 ...

**14**

votes

**1**answer

288 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\...

**8**

votes

**0**answers

80 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. ...

**5**

votes

**1**answer

394 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 ...

**3**

votes

**0**answers

124 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 ...

**1**

vote

**0**answers

60 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 ...

**1**

vote

**1**answer

58 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}....

**4**

votes

**1**answer

122 views

### 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 ...

**1**

vote

**0**answers

73 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....

**4**

votes

**1**answer

123 views

### 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 ...

**10**

votes

**1**answer

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 ...

**5**

votes

**0**answers

132 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 ...

**3**

votes

**2**answers

154 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 ...

**4**

votes

**0**answers

85 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 ...

**4**

votes

**1**answer

310 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 ...

**1**

vote

**1**answer

137 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 ...

**6**

votes

**1**answer

275 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)...

**3**

votes

**0**answers

44 views

### 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 \...

**3**

votes

**0**answers

100 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. ...

**7**

votes

**0**answers

185 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 $G$ ...

**4**

votes

**1**answer

383 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 ...

**3**

votes

**3**answers

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 ...

**2**

votes

**0**answers

102 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 ...

**3**

votes

**0**answers

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 $z,w\in G$ ...

**4**

votes

**1**answer

259 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 ...

**6**

votes

**1**answer

700 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 ...

**7**

votes

**1**answer

158 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 ...

**1**

vote

**1**answer

208 views

### cohomology ring of homogenous manifold

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :
$$
G/H ...

**1**

vote

**0**answers

134 views

### Name for the Quotient $SU(m+1)/(SU(k) \times SU(m-k))$

The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of ...

**15**

votes

**2**answers

883 views

### Is the wonderful compactification of a spherical homogeneous variety always projective?

Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}_G(H)=H$. Then by results of Brion and Pauer
...

**5**

votes

**1**answer

132 views

### Geometric interpretation of splitting of sequence associated to a homogeneous space

Let $G$ be a Lie group acting transitively on a smooth manifold $M$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\xi : \mathfrak{g} \to \Gamma(TM)$ be the Lie algebra homomorphism sending $...