# Questions tagged [symmetric-spaces]

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113
questions

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### Maximally symmetric hyperbolic 3-manifolds with finite volume

In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...

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votes

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42 views

### Closed-form expression for Riemannian exponential maps on symmetric spaces

Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...

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votes

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243 views

### Relationship between fans and root data

A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.
A toric variety is described by combinatorial information called a fan.
Both ...

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**1**answer

102 views

### What is the name of the real form corresponding to the quaternionic symmetric space?

Let $G$ be a compact simple Lie group. Choose a system of positive roots, and let $\mathrm{SU}(2) \subset G$ correspond to the highest root, and $\mathbb{Z}/2 \subset \mathrm{SU}(2)$ the centre. The ...

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181 views

### Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$

Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}_n(\mathbb{Z})$ on $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$ for each $n \leq 6$. In the ...

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52 views

### p-adic analogue of classification of irreducible Riemannian symmetric spaces?

For Riemannian symmetric spaces, we have a nice result that the simply-connected ones are products of irreducible symmetric spaces, of which we have a list of ten infinite families and some ...

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72 views

### Model geometry uniqueness

Let $ M $ be a compact connected manifold with
$$
M \cong \Gamma \backslash G /H
$$
where $G $ is a Lie group, $ H $ a compact subgroup, $\Gamma $ a discrete subgroup, and $ G/H $ is connected and ...

**4**

votes

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204 views

### Finite models for torsion-free lattices

Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?
I know this to be true in many instances (e.g. ...

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93 views

### Symmetric schemes/symmetric spaces in the category $\mathsf{Sch}_k$

From nlab, a symmetric space is a quandle object in the category of smooth manifolds. My questions are then the following:
Is there a meaningful way to define symmetric spaces in the category of ...

**6**

votes

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317 views

### Help with definition of Liouville measure

$\require{AMScd}$For a Riemannian manifold $M$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $\operatorname{T}^1(M)$ and then proceed to claim/prove that it is ...

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votes

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280 views

### Geodesic sphere in the octonion projective plane

I am considering Laplacian eigenvalues of a geodesic sphere in the octonion projective plane and I would like to know the metric on a geodesic sphere.
Does the metric on a geodesic sphere in the ...

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vote

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101 views

### Totally geodesic submanifolds of bi-invariant Lie groups

Let $G$ be a Lie group equipped with a bi-invariant metric. I have some questions concerning the totally geodesic and the flat (all sectional curvatures zero) submanifolds of $G$.
I known that every ...

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votes

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126 views

### What are the manifolds whose Curvature tensor has a globally vanishing $k$th order covariant derivative

Let $(M,g)$ be a boundaryless Riemannian manifold whose curvature tensor have the property that there exists $k\geq 2$ such that $\nabla^k R\equiv0$. What is known about such Riemannian manfiolds ? Is ...

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87 views

### Algebraic version of Loos Symmetric Space

Ottmar Loos gave a definition of symmetric spaces in terms of the existence of a multiplication map. Namely, a manifold $M$ is symmetric if there exists a multiplication morphism $\mu:M\times M\to M$ ...

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218 views

### branching laws for $p$-adic representations of reductive groups

There are many papers studying branching laws of irreducible admissible complex representations of classical groups over local fields, are there some analogues for $p$-adic representations?
For ...

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140 views

### Dependence of X in definition of Shimura variety

(Disclaimer: this question is related to this question, but is different enough that it warrants (in my opinion) a separate question)
Let $G$ be a connected reductive group over $\mathbb{Q}$. To $G$ ...

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votes

**1**answer

155 views

### Geodesic preserving diffeomorphisms of constant curvature spaces

Let $X$ be either Euclidean space $\mathbb{R}^n$, the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$.
I would like to have a classification of all diffeomorphisms $X\to X$ which map ...

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60 views

### Representation theoretic characterisation of symmetric spaces

Let $G$ be a simple compact Lie group and $H$ a closed subgroup.
Let $\mathfrak{h}\subset \mathfrak{g}$ denote the corresponding Lie algebras. Let $\mathfrak{m}$ be an orthogonal complement to $\...

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32 views

### Cylindrical coordinates in quotient of symmetric space

I am interested in the following situation. Suppose $G/K$ is a symmetric space of non-compact type and $\alpha$ is the axis of a hyperbolic isometry. I am interested in computing the Hessian of the ...

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28 views

### Discrete decomposability and asymptotic K-support

I do not know how to write the title of this question, which I think that it is a pure Lie algebra question. My question is about Example 2.14 in the paper "Discrete Decomposability of the Restriction ...

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202 views

### Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$

I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,...

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votes

**1**answer

81 views

### Homogeneity of a projective vector bundle

Let $M=G/K$ be a $G$-homogeneous manifold and suppose that $E\to G/K$ is a homogeneous (complex) vector bundle, i.e. it is defined by a representation $\phi : K \to \text{Aut(V)}$ for some complex ...

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36 views

### Embedding of Riemannian symmetric spaces $E_I$ and $E_{IV}$ into Lie group $E_6$

In answer and comments to this mathoverflow question we have discussed possiblity of embedding Riemmanian symmetric spaces $E_I, E_{II}, E_{III},E_{IV}$ of dimension $42,40,32,26$ respectively into $...

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votes

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192 views

### The automorphism group of a symplectic symmetric space

Why is the automorphism group of a sympelctic symmetric space a Lie group?
$\\$
A symplectic symmetric space is a triple $(M, \omega, s)$, where $(M, \omega)$ is a symplectic manifold and $ s \; \...

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77 views

### Maximally symmetric affine manifold

As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $n$-dimensional manifold with $\frac{n(n+1)}{2}$ Killing vectors. A ...

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206 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 ...

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66 views

### Restriction that contains a trivial representation

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$ such that $(G,G')$ is a symmetric pair. If $\pi$ is an infinitely dimensional unitary representation of $G$, ...

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135 views

### When can the metric be reconstructed (up to scaling) from knowing the conjugate points?

Let $M$ be a smooth manifold of dimension $\geq 2$. Let $g$ be a complete Riemannian metric on $M$. Let $C \subseteq M \times M$ be the set of pairs of $g$-conjugate points.
The set $C$ doesn't ...

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138 views

### Restriction of discrete series

QUESTION
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals the rank of its maximal compact subgroup. Suppose that $G'$ is a reductive subgroup of $G$ with equal rank. If $\...

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230 views

### Holonomy groups of compact Riemannian symmetric spaces

Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
https://en.wikipedia.org/wiki/...

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96 views

### What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$?

What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$? [The latter consists of those elements of the Grassmannian that can be represented by $k \times n$-matrices all of ...

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139 views

### Eigenspaces and covering relations of twisted involutions

Let $\theta:G\to G$ be an involution of a complex connected reductive Lie group, preserving a maximal torus $T$ (which, for me, lies inside a $\theta$-invariant Borel $B$). Let $K = G^\theta$ be the ...

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votes

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328 views

### Classification of compact globally symmetric spaces

It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...

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355 views

### Upper bounds for lattice points in orbits, and representations of binary quadratic forms

Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $n\geq 3$ and $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X_0\in \mathbb{Z}^{2\times n}$, ...

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259 views

### Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space

Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...

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158 views

### Orbits under an algebraic group inside a Shimura variety

Let $(G,X)$ be a Shimura datum, $K$ a compact open subgroup of $G(\mathbb A_f)$. Let $H\subset G$ be a $\mathbb Q$-subgroup. Choose a point $x\in X$. What can be said about the orbit $H(\mathbb R)\...

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71 views

### Cohomology of boundary of locally symmetric space

Let $S$ be a locally symmetric space, not necessarily compact, and $\overline{S}$ be its Borel-Serre compactification. Let $\partial S$ be the boundary of $S$. Let $\widetilde{\mathbb{C}}$ be the ...

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124 views

### Cohomology of adelic locally symmetric spaces

I am most probably wrong in asserting as follows.
Let $G$ be a connected reductive group over $\mathbb{Q}$, and $S_{K_f} = G(\mathbb{Q}) \backslash G(\mathbb{A}/K_\infty Z(\mathbb{A}) \cdot K_f $ be ...

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138 views

### Connectedness of symmetric subgroup of simply connected Lie group

Let $G$ be a connected real simple Lie group, and $\tau$ be an involutive automorphism of $G$. Then $\tau$ defines a symmetric subgroup $G^\tau$ of $G$. In general, $G^\tau$ is not necessarily ...

**12**

votes

**1**answer

259 views

### Compact quaternionic Kahler manifolds of negative curvature: examples

There is a well known problem of LeBrun-Salamon:
are there any non-symmetric compact quaternionic-Kahler
manifolds of positive scalar (and Ricci) curvature?
It is hard and still unsolved:
Quaternionic-...

**2**

votes

**1**answer

221 views

### Irreducible Symmetric Pairs

Let $\mathfrak{g}$ be a simple Lie algebra with a compact subalgebra $\mathfrak{k}$ such that $(\mathfrak{g},\mathfrak{k})$ corresponds to an irreducible Riemann symmetric space. Denote by $\sigma$ be ...

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114 views

### The uniqueness of a $K$-fixed vector in a spinor representation

Consider $G=SO(2n)$ and $K=U(n)$. $(G,K)$ is a symmetric pair. I'm interested in (zonal) spherical functions on $G/K$ which are matrix elements with respect to $K$-fixed vectors in irreducible ...

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103 views

### Symmetric pairs of holomorphic type

Let $G$ be a real simple Lie group of Hermitian type; that is, $G/K$ carries a structure of a Hermitian symmetric space where $K$ is a maximal compact subgroup of $G$. Equivalently, the center $Z(\...

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179 views

### Non-semisimple symmetric subgroups of simply connected simple algebraic groups

Let $G$ be a simply connected simple algebraic group over the field of complex numbers $\mathbb C$. Let $H$ be a symmetric subgroup of $G$. This means that there exists an automorphism of order 2 $\...

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90 views

### Isotropy symmetric holomorphic functions

Let $G$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $z\in G$.
Assume that $f$ is a holomorphic function on $G$, which is isotropy symmetric, i.e. $f\circ \varphi=f$ for any ...

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377 views

### Constructing real forms of the Tits-Freudenthal magic square for (Rosenfeld) projective planes

If $\mathbb{K},\mathbb{L} \in \{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ then the Rosenfeld projective ("elliptic"?) plane $\mathbb{P}^2(\mathbb{K}\otimes\mathbb{L})$ is "the" compact Riemannian ...

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86 views

### Two questions on homogeneous domains

Let $G$ be a domain in $\mathbb{C}^{n}$ and let $Aut(G)$ be the group of biholomorphic selfmaps of $G$. $G$ is called:
(1) homogeneous if $Aut(G)$ acts transitively on $G$, i.e. for any $z,w\in G$ ...

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122 views

### Exceptional symmetric spaces with quaternionic structure

Following this and this question I found following chain of exceptional symmetric spaces being quaternionic manifolds. I listed dimensions as superscripts for reader convenience.
$F_{I}^{28}\subset ...

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164 views

### A few questions about $E_7$ and its symmetric spaces

My question about $E_6$ survived, so I post next episode. From the Yokota book I found out that there is $-1$ in $E_7$ Lie group. This book defines Lie group $E_7$ using 56-dimensional Freudenthal ...

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153 views

### Volume growth and visibility in hyperbolic spaces

I am interested in the following two questions for complex, quaternionic and octonionic hyperbolic spaces, equipped with their usual metrics and measures:
For brevity, I will denote the volume of the ...