The symmetric-spaces tag has no usage guidance.

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### If $M$ is a globally symmetric space, do we need it to be compact to prove that all the critical submanifolds in $\Lambda M$ are nondegenerate?

In "The free loop space of globally symmetric spaces" by Ziller, he proves the following theorem:
Theorem 2. For a globally symmetric space the critical submanifolds in $\Lambda M$ are all ...

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50 views

### symmetric points on symmetric spaces

Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on ...

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29 views

### Automorphism groups of symmetric cones

Indecomposable symmetric cones fall into five classes. The automorphism group of any symmetric cone $C$ is a real Lie group $Aut(C)$. What is the associated class of Lie groups $Aut(C)$ for each of ...

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124 views

### Anosov representations and boundaries of (harmonic) maps

Let $\Sigma_g$ be a closed hyperbolic surface and $\rho\colon\pi_1\Sigma_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant ...

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88 views

### How to associate a proper parabolic subgroup of a real s.s Lie group $G$ to a non-trivial unipotent element in a non uniform lattice in $G$?

Let $G$ be a real semi-simple Lie group. Let $\Gamma$ be a non-uniform lattice in $G$.
Then it is known that $\Gamma$ contains a non-trivial unipotent element. When $\mathbb{R}$-rank of $G$ is 1, it ...

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136 views

### Non-flat totally geodesic surfaces

I'd like to know whether a Riemannian symmetric space of compact type admits a non-flat totally geodesic surface. I've found an article by Mashimo on the classification of these surfaces for certain ...

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237 views

### Volume of arithmetic quotients of symmetric spaces

Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...

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124 views

### Variation of Hodge structures associated to a hermitian symmetric domain

Let $D$ be an irreducible hermitian symmetric domain. Then there exists a variation of Hodge structures $(h_s)_{s\in D}$ on a vector space $V$ satisfying specific conditions which depend on $D$ such ...

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96 views

### Discrete subgroup of centralizer of transvections in isometries acts properly discontinuously

My question will rely on a clarification of a proof, which I simply don't understand.
Let us denote by $X$ a pseudo-riemannian symmetric space and define
$$
Z_{\mathrm{Iso}\left(X\right)}G(X) = \{\, ...

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130 views

### Kernel of the Weil homomorphism for compact symmetric spaces

Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...

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346 views

### When a Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical.
My question is When a Spherical variety is $K$-stable? Is ...

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**1**answer

250 views

### Weyl group of a symmetric space

Let $G/K$ be a symmetric space of a non-compact type, i.e. $G$ is a semi-simple connected Lie group, and $K$ is its maximal compact subgroup. Helgason in his book "Differential geometry and symmetric ...

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56 views

### Convergence of Selberg type Zeta function

Let $X=G/K$ be a Riemannian symmetric space without compact factor. We can assume $G$ is connected and real reductive. $K$ is a maximal subgroup of $G$.
Let $\Gamma$ be a discrete torsion free ...

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92 views

### Tori in Compact Riemannian Symmetric Spaces

The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic ...

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269 views

### Harish-Chandra isomorphism for compact symmetric spaces

I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
...

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94 views

### Symmetric spaces which are compact modulo the unipotent radical are compact

Is the following true?
Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$.
Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$.
Let $U$ be ...

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131 views

### Hermitian Symmetric Subspaces of Siegel Space

Let $\mathbb{H}_g$ denote Siegel space, and $M$ denote an order 4 element of the unitary subgroup $U(n)(\mathbb{R})$with $p$ eigenvalues equal to $i$, and $q$ eigenvalues equal to $-i$, $p+q=g$. ...

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59 views

### Tangent bundles of period domains of higher weight Hodge structures

Considering $A_{g}$ the moduli space of principally polarized abelian varieties, there is a variation of Hodge structures $\mathbb{V}=E^{1,0}\oplus E^{0,1}$ of weight $1$ on $A_{g}$. It is well-known ...

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256 views

### Cotangent bundle of symmetric space is symmetric space?

Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...

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370 views

### Do there exist non-totally geodesic isometric minimal immersions $\mathbb{H}^2\rightarrow G/K.$

Suppose $G$ is a non-compact, semi-simple Lie group, of rank at least two, with maximal compact subgroup $K$ and $G/K$ the corresponding Riemannian symmetric space. Let $\mathbb{H}^2(-c^2)$ be the ...

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138 views

### Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...

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### How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...

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136 views

### Cusps as warped products

It is well-known that the ends of a finite-volume hyperbolic manifold are warped products $$(0,\infty)\times_f T$$
for some euclidean manifold $T$ and $f(t)=e^{-t}$.
Question: Is there a similar ...

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146 views

### Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?

Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$.
If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial ...

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201 views

### regular semisimple elements on spherical varieties

Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$).
What can ...

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331 views

### Invariant differential operators on real Grassmannians

I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the ...

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271 views

### Equidistant hypersurfaces in symmetric space via exponentiation?

Here's some background and notation:
Let $G/K$ be a symmetric space of non-compact type. For concreteness, assume $G$ is in fact a classical simple real Lie group such as SL,SO, or Sp, and $K$ is a ...

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249 views

### Is SL_n/S(GL_k x GL_n-k) symmetric?

Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset ...

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### Seeking a generalization of group embedding of symmetric varieties

I am looking for generalizations of the following construction.
Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of ...

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101 views

### Ricci-flat Kähler metrics on symmetric varieties

Hallo,
I have a question on a paper of Azad and Kobayashi "Ricci-flat Kähler metrics on symmetric varieties". Here is the link: ...

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426 views

### Parallel forms and cohomology of symmetric spaces

Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then
$$
(\alpha \text{ is induced by an ...

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369 views

### Algebraic characterization of the curvature operator of symmetric spaces

My question is the following :
Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the ...

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279 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 ...

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357 views

### Linear symmetric spaces are spaces with ''orthogonal complements''?

The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$.
I have only recently been made aware ...

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### Chains in $K\backslash G/B$ lying over a closed $K$-orbit

Let $G$ be a complex connected reductive Lie group, $\theta$ an
involution, and $K = G^\theta$ the fixed-point subgroup.
Then $K$ has finitely many orbits on $G/B$, one of which is open
and (quite ...

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280 views

### Covering relations in $K\backslash G/B$

Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map ...

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223 views

### volume form in a symmetric space of real rank one

I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one.
The first one is the volume form induced by the Riemannian structure given by the Killing form ...

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450 views

### How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact type, $g=k+p$ the ...

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246 views

### The relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$

During my research I have recently stumbled upon the problem of finding the relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$ for $n$ large enough to be in the stable ...

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### Volume of fundamental domain and Haar measure

In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...

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### Names of noncompact riemannian symmetric spaces?

Irreducible riemannian symmetric spaces come in pairs: one compact and one not compact, usally called the noncompact dual.
The compact symmetric spaces include spheres, complex and quaternionic ...

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451 views

### different Shimura data with common underlying group?

A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...

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### Is the Baily--Borel compactification functorial?

The following question seems pretty natural, but searching online
and looking in some obvious places didn't turn up much, so maybe
I can ask it here. (Disclaimer: I'm a newcomer to this topic, so
...

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### A question about the affine Grassmanian

For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as:
$$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$
Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. ...

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### Does $\mathrm{E}_7/(\mathrm{SU}_8/(\mathbb{Z}/2))$ carry an almost complex structure?

Recall the list of irreducible simply connected inner symmetric spaces of compact type in dimension $4k+2$:
Hermitian symmetric spaces (one can write them down explicitly);
Grassmannians of oriented ...

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### links and interactions between different approaches to (super-)rigidity

By super-rigidity I mean some theorems concerning the arithmetic subgroups in semi-simple Lie groups. According to Margulis "Discrete subgroups of semi-simple Lie groups" (the book published by ...

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### Why symmetric spaces?

In Bar-Natan's "Knots at Lunch" seminar at the University of Toronto, we are currently discussing a talk by Alekseev at Montpellier about Rouvière's expansion of the Duflo isomorphism to the ...

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### question about equivariant embeddings of riemannian symmetric domains

Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a ...

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### Do symmetric spaces admit isometric embeddings as intersections of quadrics?

While preparing a seminar I gave today, the following question arose. I asked the seminar participants, but nobody knew the answer. Hence I'm asking it here in MO.
Background
Recall that a ...

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### Hermitian symmetric spaces vs Hermitian homogeneous spaces

A Hermitian symmetric space is a connected complex manifold with a hermitian metric on which the group of holomorphic isometries acts transitively, and which satisfies the following extra condition: ...