Questions tagged [symmetric-spaces]

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votes
0answers
48 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 ...
2
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1answer
151 views

Closed manifolds of nonnegative curvature operator are symmetric spaces

In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference ...
3
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0answers
99 views

Quaternonic Kähler Chern connection

For a Riemannian manifold, the natural connection is of course the Levi-Civita connection. For a complex manifold, the natural connection is the Chern connection, which coincides with the Levi-Civita ...
3
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0answers
70 views

Holonomy of a triangle in an affine symmetric space

Let $G/H$ be an affine symmetric space with involution $\sigma$, and $\mathfrak{g}=\mathfrak{m}\oplus \mathfrak{h}$ the Cartan decomposition of its Lie algebra. We can identify $G/H$ and $\exp(\...
3
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0answers
117 views

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 ...
6
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165 views

Maximal geodesic spheres in the “octooctonic projective plane”

Boris Rosenfeld claimed that the 128-dimensional compact Riemannian symmetric space on which $\mathrm{E}_8$ acts as isometries could be seen as the "octooctonionic projective plane", $(\...
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83 views

Actions of finite groups on compact symmetric spaces

I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit ...
3
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0answers
159 views

Isometry groups of symmetric spaces

Let $M=G/K$ be a symmetric space where $G=\mathrm{Isom}(M)$ and $K$ is the isotropy at some point $o\in M$. Moreover, let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the Cartan decomposition of $\...
1
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1answer
107 views

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 $\theta$...
2
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1answer
107 views

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 - ...
3
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1answer
125 views

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 ...
4
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93 views

Relationship between Hecke algebra and center of universal enveloping algebra (and the Harish-Chandra isomorphism)

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 ...
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70 views

$\tau$-admissible lift

I've been asked to take on a peer-review task which has to be completed in a short time, obviously details have to remain confidential, I need to work out what ``$\tau$-admissible lift" means if $...
3
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0answers
100 views

Anosov structure of Fuchsian representations

I was reading a paper by Labourie https://arxiv.org/pdf/math/0401230.pdf and I'm having trouble understanding a proof. First, here's the setup. $G = PSL(n,\mathbb{R})$ and $U$ is the subgroup of ...
6
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1answer
198 views

Bounded non-symmetric domains covering a compact manifold

This question is somewhat related to this other question of mine. I was wondering which are the known examples of bounded domains $\Omega$ in $\mathbb C^n$ admitting a compact free quotient. By a ...
4
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1answer
140 views

Upper bounds on the sectional curvature of the real Grassmannian

Consider the real Grassmannian as the symmetric space $\operatorname{Gr}(n,k) \cong \operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$ for $n \geq 3$, $k \geq 2$, where the metric ...
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181 views

Does cohomology ring determine a compact symmetric space?

Suppose that $M_1, M_2$ are compact connected symmetric spaces with isomorphic integer cohomology rings. Does it follow that $M_1$ is diffeomorphic to $M_2$? The only result I am aware of is this ...
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70 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 ...
6
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1answer
206 views

Involutive automorphism of simple Lie algebra

I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange. Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan ...
3
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1answer
127 views

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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0answers
47 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-...
6
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2answers
289 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 ...
5
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1answer
123 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 ...
6
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1answer
227 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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0answers
72 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 ...
3
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78 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
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2answers
217 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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0answers
97 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 ...
7
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1answer
436 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 ...
8
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2answers
352 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 ...
2
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1answer
155 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 ...
7
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1answer
140 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 ...
6
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0answers
106 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 $\mu:M\times M\to M$ ...
3
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1answer
245 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 ...
4
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0answers
150 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 $G$ ...
8
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1answer
162 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 ...
4
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0answers
77 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 $\...
2
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0answers
34 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 ...
9
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209 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,...
2
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1answer
82 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 ...
3
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0answers
38 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 $...
4
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1answer
214 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 \; \...
3
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0answers
94 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 ...
5
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0answers
223 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 ...
3
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0answers
70 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 $G$, ...
5
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1answer
147 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 ...
5
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0answers
141 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 $\...
4
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1answer
269 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/...
2
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0answers
103 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 ...
7
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0answers
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 ...