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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4 votes
2 answers
171 views

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. ...
5 votes
0 answers
77 views

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 $k$-...
1 vote
1 answer
118 views

how to construct a finite energy map

In the construction of harmonic maps by Eells and Sampson, one needs to start with a map with finite energy and use the heat equation to deform it into a harmonic map. The construction of such a ...
1 vote
0 answers
57 views

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 $...
8 votes
2 answers
342 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 ...
2 votes
0 answers
151 views

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) ...
8 votes
1 answer
493 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. ...
4 votes
0 answers
134 views

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 ...
4 votes
0 answers
270 views

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}} ...
2 votes
1 answer
308 views

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 ...
1 vote
2 answers
154 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....
4 votes
2 answers
334 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 ...
9 votes
2 answers
316 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 ...
1 vote
0 answers
27 views

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 ...
1 vote
0 answers
29 views

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?
5 votes
1 answer
304 views

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 ...
1 vote
0 answers
82 views

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 ...
6 votes
1 answer
464 views

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 ...
3 votes
1 answer
201 views

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 $...
8 votes
1 answer
375 views

What kind of locally symmetric space is a rational sphere

Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere. My question is the following. Is there other ...
2 votes
0 answers
47 views

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 ...
1 vote
0 answers
129 views

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 ...
6 votes
0 answers
140 views

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 $. ...
5 votes
1 answer
145 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 ...
2 votes
2 answers
451 views

Combination theorems for discrete subgroups of isometry groups

Maskit's combination theorem says: if $M=M_1\cup_\Sigma M_2$ is a union of hyperbolic 3-manifolds $M_1=\Gamma_1\backslash H^3, M_2=\Gamma_2\backslash H^3$ along a surface $\Sigma$, and if the limit ...
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?...
1 vote
0 answers
59 views

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 ...
5 votes
1 answer
180 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 ...
12 votes
0 answers
223 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: ...
2 votes
2 answers
315 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 ...
2 votes
1 answer
273 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 ...
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 ...
21 votes
2 answers
1k views

Geometric interpretation of exceptional symmetric spaces

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
1 vote
1 answer
191 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 ...
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 ...
2 votes
0 answers
66 views

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{...
3 votes
0 answers
105 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_{...
1 vote
1 answer
116 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 $...
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,...
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 ...
7 votes
1 answer
213 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 $...
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 $0$....
3 votes
1 answer
242 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 ...
5 votes
0 answers
250 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(...
4 votes
1 answer
118 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,...
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}(\...
5 votes
1 answer
155 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 ...
3 votes
0 answers
54 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 ...
8 votes
1 answer
240 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 ...
9 votes
0 answers
469 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 ...