Questions tagged [homogeneous-spaces]
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251 questions
6
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2
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Trivialising homogeneous vector bundles over a homogeneous space
Take the sphere $S^2$ with the standard indexing of its line bundles $E_k$, for $k$ an integer. Is it true that $E_{k} \oplus E_{-k}$ is a trivial vector bundle? If so, what is the easiest way to see ...
- 267
1
vote
0
answers
121
views
Confusion about the definition of homogeneous orbits from Ratner's theorem
I was reading Ratner's Raghunathan’s topological conjecture and distributions of unipotent flows and confused about a definition in the first page: Ratner used the right action $\Gamma \backslash G$ ...
- 495
2
votes
0
answers
59
views
Comparing $i$ th minima of a lattice and discrete subgroup
This is a continuation of the previous question Comparison between first minimum of a lattice and a discrete subgroup in function field. Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}...
- 151
6
votes
1
answer
465
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Seeking example of Dynkin index 1 Lie subalgebra with specific properties
$\def\h{\mathfrak{h}}\def\g{\mathfrak{g}}\def\m{\mathfrak{m}}$ I am looking for an example, a class of examples, or a proof that none exist [ADDED 11 July '25: See below, for why I now think this is ...
- 37k
5
votes
1
answer
331
views
Question about Kapranov's resolution of the diagonal for Grassmannians
I have been thinking about derived categories of homogeneous spaces a little bit lately, especially $\mathrm{D}^{\mathrm{b}}(\mathrm{Gr}(k,V))$. Some foundational results are proven in a paper of ...
- 403
1
vote
0
answers
62
views
O(3)/SO(3) as O(2)^- complex
I'm totally new to equivariant topology. I would like to ask what is the decomposition of orbit type when we treat $O(3)/SO(3)$ as a $O(2)^-$ complex. Here $O(2)^-$ is the twisted subgroup of $O(3)$, ...
- 11
5
votes
0
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164
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Equivalent definitions of volume of representations (or characteristic classes of flat bundles)
Statement of the problem:
Let $M$ be a closed connected oriented manifold with fundamental group $\Gamma$ (considered as covering transformations acting on its universal cover). Let $G$ be a (...
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7
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1
answer
254
views
Forms of Grassmannians
The question is complementary to this question. How to construct pointless real forms of Grassmannians $\operatorname{Gr}(k,n)$? For which $k$ and $n$ do they exist? Any reference will be very helpful....
1
vote
1
answer
231
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Comparison between first minimum of a lattice and a discrete subgroup in function field
This is the continuation of the previous question Lattices in the extension field of local fields in positive characteristic. Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}_q$, where ...
- 151
4
votes
1
answer
303
views
Is $\mathbb H^{p,q} _\mathbb C$ a symmetric space?
I asked a similar question yesterday and got negative votes. I was thinking I should ask the question in more details. I am very new to symmetric space stuff and if anyone knows this, please advise....
- 185
2
votes
1
answer
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Given a metric space $M$ and $a,b \in M$, does there exist a compatible metric $d$ under which $d(a,c)+d(c,b) = d(a,b)$ for all $c \in M$?
Every separable metric space $M$ embeds homeomorphically in the Hilbert cube $H = [0,1]^\omega$. Since the cube is $2$-homogeneous (indeed $n$-homogeneous for any $n$) we can assume any two given ...
- 2,085
11
votes
0
answers
474
views
Have you seen this invariant 3-form?
I've calculated a particular invariant 3-form on a reductive homogeneous space $G/H$ that is associated to an invariant principal connection $A$ on $G\to G/H$, and I want to know if it is something ...
- 37k
3
votes
1
answer
368
views
Lattices in the extension field of local fields in positive characteristic
Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}_q$, where $q$ is a prime power. The norm in this field is defined by
$
\left| \frac{f}{g} \right| = q^{\deg(f) - \deg(g)},
$
where $f, ...
- 151
0
votes
1
answer
253
views
Smooth manifolds with minimally transitive diffeomorphism group
I know that there are vertex-transitive graphs, $\Gamma$, whose automorphism group, $\text{Aut}(\Gamma)$, is minimally transitive. That is, $\text{Aut}(\Gamma)$ acts transitively on $\Gamma$ but no ...
- 141
1
vote
0
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61
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Example of a smooth 4-manifold $M_3 \times \mathbb{R}$ which is not homogeneous w.r.t. any Lie group [duplicate]
I am looking for an example of a smooth 4-manifold $M_4$ which is not homogeneous with respect to any finite dimensional Lie group. For physics-related reasons, I am hoping to find a case where $M_4 = ...
- 141
2
votes
0
answers
361
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De Rham cohomology of a $7$-dimensional compact locally homogeneous manifold
I am trying to compute the de Rham cohomology of the 7-dimensional compact smooth manifold $X:=\Gamma\backslash G/K$ given by the quotient of the homogeneous space
$$
G/K:=(\text{SL}(2;\mathbb{C})\...
- 31
2
votes
0
answers
77
views
Are maps between cohomology of homogeneous vector bundles morphisms of representations?
Let $X = G/P$ a rational homogeneous variety, e.g. a grassmannian. Consider a short exact sequence $$ 0 \longrightarrow E_1 \longrightarrow E_2 \longrightarrow E_3 \longrightarrow 0$$
where $E_i$ are ...
- 41
3
votes
1
answer
211
views
For $\mathbb R^n \times Q \cong \mathbb R^m \times Q $ must $n = m$? ($Q$ is the Hilbert cube)
There are several theorems describing the topology on hyperspaces of convex subsets of $\mathbb R^n$ under the Hausdorff metric. For example Antonyan and Jonard-Pérez prove the space of compact convex ...
- 2,085
1
vote
1
answer
285
views
Determinant bundle over homogeneous varieties
I am looking for a way to compute the determinant of a homogeneous vector bundle over any homogeous variety. I am aware of how these computations work for the $A_n$ case (i.e., for flag varieties), ...
- 41
5
votes
2
answers
397
views
Integral points on homogeneous spaces over a DVR
Let $R$ be a DVR (possibly mixed characteristic) with fraction field $K$. Let $V \to \operatorname{Spec} R$ be a smooth affine scheme with a transitive action of $GL_{n,R}$ so that each geometric ...
- 3,680
9
votes
3
answers
946
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A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \...
1
vote
1
answer
167
views
How to show the geodesic orbit of a badly approximable number are/are not homogeneously equidistributed on its orbit closure?
Let $x\in \mathbb R$ be a badly approximable number. By Dani's correspondence, $g_t u_x \mathbb Z^2$ is bounded away from the cusp in $X=\text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$, identified with ...
- 495
3
votes
0
answers
78
views
Algebraicity of the group of equivariant automorphisms of an almost homogeneous variety
The base field is the field of complex numbers. Let $G$ be a connected linear algebraic group. Let $X$ be an almost homogeneous algebraic variety, i.e. $G$ acts on $X$ with a dense open orbit $U \...
- 585
2
votes
0
answers
83
views
De Rham product decomposition theorem in a particular setting
Let $G$ be a Lie group, and $H$ a Lie subgroup of $G$ such that $G/H\sim \mathcal M$ is a homogeneous space diffeomorphic to $\mathbb{R}^n$, equipped with an invariant Riemannian metric $g$. If this ...
- 401
2
votes
1
answer
336
views
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 ...
- 317
2
votes
0
answers
152
views
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
$...
4
votes
1
answer
250
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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 ...
- 235
0
votes
1
answer
218
views
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 ...
- 347
7
votes
1
answer
212
views
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 $\mathscr{B}$...
- 73
6
votes
0
answers
381
views
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 (...
- 4,577
4
votes
0
answers
273
views
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 ...
- 4,577
8
votes
0
answers
267
views
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 \...
- 895
4
votes
1
answer
479
views
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–...
- 321
1
vote
2
answers
244
views
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 ...
- 1,065
3
votes
1
answer
559
views
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 $J$, ...
- 3,878
3
votes
0
answers
102
views
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 ...
- 2,577
8
votes
2
answers
720
views
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 ...
- 553
2
votes
1
answer
355
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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.
- 343
3
votes
1
answer
158
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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 ...
- 321
3
votes
0
answers
193
views
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 ...
- 321
4
votes
0
answers
142
views
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 ...
- 37.2k
5
votes
0
answers
188
views
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 ...
- 325
2
votes
0
answers
178
views
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\...
- 495
2
votes
0
answers
216
views
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 ...
- 895
5
votes
2
answers
434
views
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. ...
- 895
5
votes
1
answer
323
views
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 ...
2
votes
1
answer
426
views
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): ...
- 3,878
3
votes
0
answers
69
views
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 ...
- 5,377
7
votes
1
answer
356
views
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_{\...
- 267
1
vote
0
answers
105
views
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 ...
- 495