# 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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### 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 ...
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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 ...
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### Multiplicative invariants of non-reduced root systems

It is a well known fact (cf.  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 ...
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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 ...
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### 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 ...
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### 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?...
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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 ...
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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 $M$. ...
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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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### 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: ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Functoriality for compactifications of locally symmetric spaces

Let $X$ be a symmetric space associated to an algebraic group $G$ defined over $\mathbb{Q}$ and $G(\mathbb{R})$ acts on $X$ from the left. Let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup ...
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### Existence of commuting Chevalley involution

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra, and let $\theta$ be a complex linear involution on $\mathfrak{g}$. Let $\mathfrak{a}$ be a Cartan subspace, and choose a $\theta$...
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### Automorphism group of Hermitian symmetric spaces

For a hermitian symmetric space $M$ one has its group of biholomorphic maps $\operatorname{Hol}(M)$ and its group of Riemannian isometries $\operatorname{Isom}(M)$. According to Prop. 1.6 of Milne - ...
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### A different notion of a decomposable symmetric tensor (besides Veronese)

$\DeclareMathOperator{\complex}{\mathbb{C}}$ Let $\bigvee^m(\complex^n)\subseteq (\complex^n)^{\otimes m}$ denote the space of symmetric tensors, i.e. the set of $x \in (\complex^n)^{\otimes m}$ that ...
Let $G$ be a semisimple Lie group of noncompact type and let $K$ be a maximal compact subgroup. Let $\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{k}$ be the Cartan decomposition coming from some ...