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.
182 questions
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Central isogeny, Shimura varieties and exceptional cases
For a simple complex Lie algebra $\mathfrak g$, its weight lattice is not equal to the root lattice (i.e. the center of its simply connected form is a non-trivial finite group) iff $\mathfrak g$ is of ...
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Analytic and Algebraic Brauer groups of complex symmetric spaces
Let $G$ be a connected reductive group and $K=G^{\sigma}$ where $\sigma$ is an involution of $G$. Could someone please help me to understand what will be the (Analytic) Algebraic Brauer groups of the ...
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Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
3
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1
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Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
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Why automorphism group of a Hermitian symmetric domain has trivial center?
Definition: A Hermitian symmetric domain is a Hermitian manifold that is connected, homogeneous, has a symmetry at some point (by homogenity hence every point), and has negative curvature.
I want to ...
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45
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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 ...
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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) ...
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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 ...
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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 ...
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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....
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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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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}}
...
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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 $...
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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 ...
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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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0
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96
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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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74
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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
$$Y_K := G(\mathbb{...
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Almost two-point homogeneous spaces
I define a NICE space to be a connected Riemannian manifold $M$ such that for any two distinct points $p,q\in M$, there exists an isometry $R_{p,q}$ exchanging these two points (that is such that $R_{...
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Distribution of top left block from unitary symmetric matrices
If $U$ is a $N\times N$ random unitary matrix uniformly distributed with respect to Haar measure, a $M\times M$ block $A$ from it has distribution given by
$$ \det(1-AA^\dagger)^{N-2M}.$$
If $O$ is a $...
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How to construct lattice points in bounded symmetric domain?
Consider the Hermitian bounded symmetric domain for $k \leq m$:
$$
C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \}
$$
where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken,...
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Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?
In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
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Invariants for the isotropy representation of a Riemannian symmetric space
Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $...
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General fiber and the symmetric product of an ample hypersurface
Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
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Classification of "homogeneous" submanifolds of ℝⁿ
I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean manifold" if:
it is a closed connected smooth submanifold of $\mathbb R^n$,
for every $p, q$ in $M$, there is a ...
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Fundamental group of compact globally symmetric spaces
The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient
$$(*) \quad \pi_1(...
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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,...
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85
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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}(\...
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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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59
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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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247
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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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