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Questions tagged [symmetric-spaces]

A symmetric space is a connected Riemannian manifold in which at every point there exists a global self-isometry whose differential at the given point is minus identity.

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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 ...
Chevallier's user avatar
4 votes
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References for $K$-orbits in $G/B$

Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
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Spherical functions in the space of functions on real Grassmannians

Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$. Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
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Harmonic analysis of vector bundles on symmetric spaces

This is a follow-up to my previous question. Given a semisimple symmetric space $M\simeq G/H$, in particular, the real hyperbolic space $H_{p,q}\simeq SO(p,q)/SO(p,q-1)$, and a vector bundle $E$ over $...
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A property of the Riemannian metric on the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$

Consider the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$. For any $M\in X_d$, by transporting the Killing form on $T_I(X_d)$ (the space of symmetric matrices with trace zero) ...
Random's user avatar
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Relation between two Harish-Chandra homomorphisms

Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
S. D. Z's user avatar
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How to prove Siegel upper half plane is a hermitian symmetric space

There is a statement that is Siegel upper half plane of genus g, $\mathbb{H}_g:=\left\{Z=X+i Y \in M_n(\mathbb{C}) \mid X, Y \text { real }, Z=Z^{T}, Y=\operatorname{Im} Z>0\right)$ is isomorphic ...
AlphaNotKnows's user avatar
1 vote
2 answers
160 views

Classification of compact isotropy irreducible homogeneous Kaehler manifolds

Is classification of compact isotropy irreducible homogeneous Kaehler(-Einstein) manifolds known? Here, a homogeneous space is called isotropy irreducible if the isotropy representation is irreducible....
Castle's user avatar
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Bi-$M$-invariant measure on a Riemannian symmetric space

Let $G$ be a noncompact connected semi simple Lie group. Let $K$ be a maximal compact subgroup and $G=K\overline{A_{+}}K$ be a Cartan decomposition of $G.$ Let $M=Z_{K}(\mathfrak{a})$. Then how to ...
A beginner mathmatician's user avatar
8 votes
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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 ...
G. Gallego's user avatar
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Hermitian locally symmetric space with nonnegative bisectional curvature

Let $(M,g)$ be an Hermitian locally symmetric space with nonnegative bisectional curvature. Suppose the fundamental group of $M$ is finite, can we prove that $M$ is simply-connected?
Zhiqiang's user avatar
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9 votes
2 answers
322 views

Almgren's regularity Theorem ; a simple example?

Let me remind Almgren's regularity Theorem: the singular set of area-minimizing surface has codimension at least $2$. I wish to share here a simple example in low dimension, although I don't know ...
Denis Serre's user avatar
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Buildings as generalizations of symmetric spaces

In almost every introductory notes on Tits buildings these are motivated as structures capturing/ sharing several features of symmetric spaces. Could somebody elaborate what are precisely the main ...
user267839's user avatar
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Symmetric cones and symmetric spaces

I start by stating what I think I understood on symmetric cones (https://en.wikipedia.org/wiki/Symmetric_cone). Let $\mathcal{C}$ be a symmetric cone in a vector space $V$. There are Riemannian ...
Chevallier's user avatar
6 votes
1 answer
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Definition of locally symmetric space of reductive groups

This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless. In my attempt to study Shimura varieties, I came across ...
Coherent Sheaf's user avatar
4 votes
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The convention of Fourier transform on symmetric spaces

When trying to understand the Plancherel formula of reductive symmetric space of Harish-Chandra class, I get confused on the convention of Fourier and related transforms. $\newcommand{\H}{\mathcal{H}} ...
Lacia's user avatar
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3 votes
1 answer
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Cut locus for simply connected manifolds

Let $M$ be a connected and simply connected compact Riemannian manifold (without boundaries). Fix a point $p\in M$. The diameter set $D_p$ of $p$ is the set of points that maximize the distance from $...
Lau's user avatar
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2 votes
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Multiplicative invariants of non-reduced root systems

It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...
G. Gallego's user avatar
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A question about Araki's 1962 paper on classification of irreducible symmetric spaces

I am looking at Sôhô Araki's 1962 paper for the classification of real semisimple lie algebras. Here's the link to the paper: On root systems and an infinitesimal classification of irreducible ...
Soumyadip Sarkar's user avatar
5 votes
1 answer
152 views

Maximum symmetry metric on Cayley plane $ F_4/{\operatorname{Spin}(9)}$

The maximum symmetry metric on real projective space $ \mathbb{RP}^n $ is the round metric. The maximum symmetry metric on complex projective space $ \mathbb{CP}^n $ is the Fubini–Study metric; see ...
Ian Gershon Teixeira's user avatar
11 votes
2 answers
1k views

Is there a contractible hyperbolic 3-orbifold of finite volume?

Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$. Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that \begin{equation} X:=\mathbb{H}^3/\Gamma \end{equation} is contractible?...
David.D's user avatar
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A generalised spectral theorem for symmetric *spaces* generalising the one for symmetric matrices

The set of $n \times n$ symmetric matrices over $\mathbb R$ form a symmetric space. The relevant Lie group is $G = GL_n(\mathbb R)$ and the relevant involution is $\sigma(X)=X^{-T}$; it follows then ...
wlad's user avatar
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6 votes
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Maximum symmetry metric on irreducible compact symmetric space

Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
181 views

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 ...
SophieS's user avatar
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12 votes
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226 views

Symmetric spaces are quandles. Is this important?

For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
Spencer Leslie's user avatar
3 votes
0 answers
134 views

Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces

Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
Callum's user avatar
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2 votes
2 answers
319 views

Are the automorphism groups of simple symmetric cones algebraic groups?

This question arises when I tried to understand Chapter 2 of the celebrated book "Smooth compactification of locally symmetric varieties" by Ash–Mumford–Rapoport–Tai. The setting is as ...
Mingchen Xia's user avatar
8 votes
1 answer
506 views

Explicit construction of a (the?) dual symmetric space

I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type. ...
S.T.'s user avatar
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1 answer
193 views

Semisimple Lie algebra and convexity

There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...
Nicolas Medina Sanchez's user avatar
1 vote
0 answers
93 views

Locally symmetric spaces dependence on number field

A special case of locally symmetric spaces is the moduli space of abelian varieties of a given dimension $g$ (over a given base field $k$), lets call it $\mathcal{A}_g$ and is a $k$-scheme. For any ...
curious math guy's user avatar
2 votes
1 answer
277 views

Complex quadric as a symmetric space

It is known that a smooth complex quadric is a symmetric space. For example, it is $$\operatorname{Spin}(n+2)/G$$ where $G$ is the maximal parabolic subgroup. I want a reference for more details and ...
Peng's user avatar
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2 votes
0 answers
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Image of tori in locally symmetric spaces and homology

Suppose we have a reductive group $G$ over $\mathbb{Q}$, a compact subgroup of the adelic points $K_f\subset G(\mathbb{A}_{\mathbb{Q}})$, and the associated locally symmetric space $$Y_K := G(\mathbb{...
xir's user avatar
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3 votes
0 answers
107 views

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_{...
Andrea Aveni's user avatar
1 vote
1 answer
117 views

Distribution of top left block from unitary symmetric matrices

If $U$ is a $N\times N$ random unitary matrix uniformly distributed with respect to Haar measure, a $M\times M$ block $A$ from it has distribution given by $$ \det(1-AA^\dagger)^{N-2M}.$$ If $O$ is a $...
Marcel's user avatar
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2 votes
0 answers
43 views

How to construct lattice points in bounded symmetric domain?

Consider the Hermitian bounded symmetric domain for $k \leq m$: $$ C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \} $$ where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken,...
Vít Tuček's user avatar
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12 votes
1 answer
298 views

Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
Zhenhua Liu's user avatar
7 votes
1 answer
218 views

Invariants for the isotropy representation of a Riemannian symmetric space

Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $...
Gro-Tsen's user avatar
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0 votes
0 answers
75 views

General fiber and the symmetric product of an ample hypersurface

Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
Roxana's user avatar
  • 519
3 votes
1 answer
245 views

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 ...
Andrea Aveni's user avatar
5 votes
0 answers
257 views

Fundamental group of compact globally symmetric spaces

The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient $$(*) \quad \pi_1(...
Lucas Seco's user avatar
  • 1,113
4 votes
1 answer
119 views

Rigidity of the compact irreducible symmetric space

Let $(M^n,g)$ be an irreducible symmetric space of compact type. In particular, $(M^n,g)$ is an Einstein manifold with a positive Einstein constant. Is there any classification for $(M^n,g)$ if $(M^n,...
Zhiqiang's user avatar
  • 881
3 votes
0 answers
76 views

Real algebraic structure on locally symmetric varieties

Let $\mathbf{G}$ be a semisimple $\mathbb{Q}$-algebraic group, $\mathbf{G}(\mathbb{R})^+$ the connected component (for the Euclidean topology) containing the identity, $\Gamma\subset \mathbf{G}(\...
Chris's user avatar
  • 786
5 votes
1 answer
163 views

Proof of equivalence between Lie triple systems and totally geodesic submanifolds

In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a ...
Matteo Raffaelli's user avatar
3 votes
0 answers
56 views

Examples of non-affine, non-semisimple symmetric spaces

I am looking for examples of pseudo-Riemannian symmetric spaces that are not of the type encountered in the standard Riemannian classification, i.e. not flat or with semisimple symmetry group. As I ...
Stijn's user avatar
  • 131
8 votes
1 answer
242 views

Covolumes of unit groups of division algebras

Let $D$ be a central division (or maybe just simple) algebra over $\mathbb{Q}$. Let $\mathcal{O} \subset \mathcal{O}_m$ be an order inside a fixed maximal order and denote by $\mathcal{O}^1$ its group ...
Radu T's user avatar
  • 767
9 votes
0 answers
480 views

Volume of a geodesic ball in $\operatorname{SL}(n) / {\operatorname{SO}(n)}$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son Question: What is the volume of a ...
ccriscitiello's user avatar
4 votes
2 answers
343 views

Correct curvature tensor of symmetric space of positive definite matrices with trace metric?

Let $Pos(n)$ be the set of $n \times n$ real positive definite matrices with trace (aka affine-invariant) metric $$\langle u, v \rangle_p = tr(p^{-1} u p^{-1} v)$$ for all $p \in Pos(n)$ and $u, v \in ...
ccriscitiello's user avatar
5 votes
2 answers
269 views

Jacobi fields on non-symmetric spaces

I am searching for examples of manifolds which are not symmetric spaces but where Jacobi fields can be computed in closed form. For now, I am aware of Gaussian distribution with the Wasserstein ...
Chevallier's user avatar
2 votes
0 answers
81 views

Showing an action of a higher rank lattice on hyperbolic space has a fixed point

In the introduction to this paper, the author mentions that any action of a lattice $\Gamma < G$ on a rank one symmetric space $X$ has a fixed point, where $G$ is a higher rank semisimple algebraic ...
Mauro's user avatar
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3 votes
0 answers
92 views

Is the affine Grassmannian manifold a symmetric homogeneous space?

I am interested in the manifold of affine subspaces of dimension $k$ of $\mathbb{R}^n$, which can be viewed as the homogeneous space $$ E(n)/(E(k)\times O(n-k)),$$ where $E$ refers to rigid motions ...
Chevallier's user avatar