# Questions tagged [homogeneous-spaces]

The homogeneous-spaces tag has no usage guidance.

231
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### Mulitplicity one property for $\mathcal{D}'$ and $L^2$ over a homogeneous space

Let $G$ and $G_0$ be Lie groups, and suppose that a homogeneous space $X=G/G_0$ have a $G$-invariant measure.
It is known (E.G.F. Thomas showed) that there is an admissible parametrization $\{\mathcal{...

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115
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### Connecting homomorphism in non-abelian cohomology

Let $G$ be a simply connected, semisimple algebraic group over $\mathbb{R}$ and let $X$ be a homogeneous space for $G$ with finite commutative stabilizer $\mu$. There is a connecting homomorphism from ...

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### The double quotient of SU(N) by its diagonal maximal torus

$\DeclareMathOperator\SU{SU}$The special unitary group $\SU(N)$ contains $T^{N-1}$ as a maximal torus, which we take to be the diagonal subgroup of $\SU(N)$. Can we describe the double quotient space
$...

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### Curvature calculation of $S^{2n+1}=U(n+1)/U(n)$ as a homogeneous space

I asked this question at StackExchange, but got no answer. So I am reposting it here.
I eventually want to check how the Hopf fibration
$$
S^{2n+1}\to {\mathbb C}P^n
$$
satisfies the Riemannian ...

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### Help in understanding the singular system of linear forms and non escape of mass

I am having some trouble in understanding certain portions of the following paper by KKLM
https://link.springer.com/article/10.1007/s11854-017-0033-4
So in proposition 3.1, they proved the estimate ...

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### Criteria on $f$ such that $\begin{bmatrix} 1 & f(t)\\ 0 & 1 \end{bmatrix}\mathbb Z^2$ is equidistributed on the circle (periodic orbit)

Consider $\left(\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}\mathbb Z^2 \right)_{t\in \mathbb R} \cong S^1$. Let $\mu$ denote the rotation invariant Haar measure $m$ on this orbit. I wonder if ...

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### Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...

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342
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### Why can't a Lie group act transitively on a finite volume hyperbolic manifold?

In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...

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139
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### Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?

Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...

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### Lattice point counts on the determinantal variety

I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$.
$\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \...

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### Equidistribution of the orbit $\{\text{diag}(t^a,t^{-a})\Lambda \}_{t>0}$ for a.e. $\Lambda\in \text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$

$\DeclareMathOperator\diag{diag}\DeclareMathOperator\SL{SL}$It is well-known that geodesic flow $g_t=\{\diag(e^t,e^{-t}) \}_{t>0}$ acts ergodically (actually mixing) on $\SL(2,\mathbb R)$ (Howe–...

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### Generalization of the flatness of $\mathbb R^3$

Original setup
Consider the manifold $M=\mathbb R^3$ with the natural vector bundle connection $\nabla$. This connection, like any connection on a vector bundle, induces, or is induced by, a principal ...

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269
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### Mackey coset decomposition formula

I have a question about following argument I found
in these notes on Mackey functors:
(2.1) LEMMA. (page 6) Let $G$ be a finite group and $J$ any subgroup. Whenever $H$ and $K$ are subgroups of $J$, ...

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### Can a semisimple orbit always be identified with a cotangent bundle?

Let $H$ be a semisimple element of the Lie algebra $\mathfrak{g}$ of a semisimple Lie group $G$, and let $M:=\mathrm{Ad}_G(H)\subset\mathfrak{g}$ be the corresponding adjoint orbit. If we choose a ...

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### Spherical roots, restricted roots, and the dual group of a symmetric variety

Let $k$ be an algebraically closed field of characteristic $0$ and $G$ a semisimple simply-connected group over $k$. Consider a symmetric variety of the form $X=G/H$, for $H=G^\theta$ the fixed point ...

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### Fourier transforms of homogeneous functions [closed]

Compute Fourier transforms of homogeneous functions of the form,
$$
\frac{1}{|x|^{n+d}}P_d(x)
$$
where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.

3
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1
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114
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### Existence of a bounded measurable subset of $\text{SL}(d,\mathbb R)$ that is Borel isomorphic to $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$?

$\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$ has two interesting properties: on one hand it is non-compact, but on the other hand it admits a unique $\text{SL}(d,\mathbb R)$-invariant finite ...

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### Integrating over a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action and the choice of the fundamental domain

Let $\mathcal{F}$ be a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action. It is well known that there exists a unique $\text{SL}(d,\mathbb R)$-invariant probability ...

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### Monodromy action on homogeneous spaces

If $H$ is a Lie subgroup of $G$, then there is a fibration sequence
$$
G/H\to BH\to BG.
$$
By choosing a model for $EG$ we can promote this into a fibre bundle.
My question is about how to understand ...

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### Is every linear Lie group of bounded geometry?

$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...

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### Double quotient integral formula on $\Gamma \backslash G /K$

Let $G=\text{SL}(n,\mathbb R)$, $\Gamma=\text{SL}(n,\mathbb Z)$ and $K=\text{SO}(n,\mathbb R)$. Consider the double coset space $X= \Gamma \backslash G /K$ and its fundamental domain $\mathcal F\...

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156
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### Weil's "Sur la formule de Siegel dans la théorie des groupes classiques"

I am struggling to understand the following passage on Weil's text "Sur la formule de Siegel dans la théorie des groupes classiques" on page 16.
If I understood correctly, in the second ...

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345
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### Integrating on orbits of algebraic groups

Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. ...

5
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219
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### Non-integrable almost complex structure for complex projective $3$-space

It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this article for ...

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278
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### Product decomposition into semisimple and unipotent parts of an algebraic group (in Borel’s LAG)

Let $G$ be an algebraic group, i.e., an affine reduced, separated $k$-scheme of finite type with structure of a group. In Borel’s Linear Algebraic Groups Theorem III.10.6(4) says
Theorem 10.6 (3): ...

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### Are there invariants of configurations of points in space obtainable via the moduli space of solutions of the Berry-Robbins problem?

Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the ...

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### Non-homogeneous line bundles over a homogeneous space

Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form
$$
G \times_{\...

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### Second moment version of the multiple-sum Rogers integration formula

I know the following theorems due to Rogers. Let $X$ denote the space of $n$-dimensional unimodualar lattices in $\mathbb R^n$, equipped with the canonical Haar measure.
Theorem 1(Siegel-Rogers). Let ...

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### Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds

I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...

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### Free $S^1$-action on compact homogeneous spaces

Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
If $r(G) > r(K)$ (...

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### How large is the set of unimodular lattices whose sucesssive minima cannot be attained by a basis of lattice?

Recall that the $i$-th successive minimum of $L\in \mathcal L$ (space of full rank lattices in $\mathbb R^d$), denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...

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### Can all hermitian symmetric spaces be realised as coadjoint orbits?

Here is what I know. Assume $M\cong G/K$ is an irreducible hermitian symmetric space. Denote the Lie-algebra of $K$ by $\mathfrak{t}$. Proposition 1.2. chapter 3 in
Wienhard - Bounded cohomology and ...

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### m-point-homogeneous, but not (m+1)-point-homogeneous

It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large).
In other words, if $A\subset Q$ ...

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### Exists $G$-equivariant embedding with faithful representation of $G$?

Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...

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### Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic

I wonder if there are any direct proof that $g_t=\operatorname{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic (or even stronger,...

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### Coinvariants of a homogeneous space

We work over $\mathbb{C}$. Let $G$ be an algebraic group, and let $X$ be an affine homogeneous $G$-variety. Write $\mathbb{C}[X]$ for the algebra of regular functions on $X$, which is itself a $G$-...

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### About the classification of simply connected homogeneous 3-manifolds

I've read somewhere (but cannot locate the source) that the following classification holds: simply connected homogeneous 3-manifolds are either isometric to $S^2 \times \mathbb{R}$ or to a metric Lie ...

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### Do all homogeneous spaces have homogeneous compactifications?

Let $X$ be a separable metric space which is homogeneous, i.e. for every two points $x,y\in X$ there is a homeomorphism $h$ of $X$ onto itself such that $h(x)=y$.
A compactification of $X$ is a ...

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### Almost two-point homogeneous spaces

I define a NICE space to be a connected Riemannian manifold $M$ such that for any two distinct points $p,q\in M$, there exists an isometry $R_{p,q}$ exchanging these two points (that is such that $R_{...

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### Is this quotient of $\mathbb{C}^{m+1}$ by $U(1)$ only "nice" for $m=1$?

Let $V^{m+1} = \mathbb{C}^{m+1}$ and let $U(1)$ act on it by its diagonal representation, so that really, it is just like scalar multiplication by a unit modulus complex number.
I am interested in the ...

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### Scalar curvature of homogeneous bounded domains

Is it true that a bounded domain $\Omega \subset \mathbb C^n$ equipped with a homogeneous metric (not necessarily a Bergman metric) has (constant) negative scalar curvature?

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### Homogeneous metrics on compact Lie groups

Given a differentiable manifold $M$, a Riemannian metric $g$ on $M$ is called homogeneous if its isometry group acts transitively on $M$. In that case, given any group $G$ acting transtitively on $(M,...

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### Intersection of open orbits in homogeneous space

Let $G$ be a simple complex algebraic group. Let $P(\alpha_i),P(\alpha_k)$ be maximal standard parabolic subgroups of $G$ associated to simple roots $\alpha_i,\alpha_k$ in the root system associated ...

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### The (last step of the) proof that the set of badly approximable matrices has measure zero

An $m \times n$ matrix $A$ is called badly approximable if there exists $c > 0$ such that for all integer vectors $p \in \mathbb Z^m$ and $q \in \mathbb Z^n-\{0\}$ we have
$$ \|A q + p \| \ge c \| ...

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### Classification of "homogeneous" submanifolds of ℝⁿ

I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean manifold" if:
it is a closed connected smooth submanifold of $\mathbb R^n$,
for every $p, q$ in $M$, there is a ...

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### Classification of homogeneous Einstein manifolds

In Besse's "Einstein manifolds", p. 177, he states that, until that moment, no general classification of homogeneous Einstein manifolds was know, even in the compact case. More specifically, ...

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### An analogy of product formula for homogeneous space?

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)...

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### Decomposition of fiber product of $G$-sets in $G$-orbits

I have posted an identical question in MSE few days ago, but maybe this site is a better adress to discuss this problem:
Let $G$ be a finite group and $K, H \leq G$ two subgroups. Then
the right ...

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### Homogeneous space and rational section

Let's embed $\operatorname{SO}_n$ inside $\operatorname{GL}_n$ through the standard representation. Does the map $\operatorname{GL}_n\rightarrow \operatorname{GL}_n/{\operatorname{SO}_n}$ admit a ...

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### Properties of the orbit $Abx_0$ when $b$ is upper or lower triangular but not diagonal

Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $A$ denote the subgroup of $G$ ...