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9
votes
3answers
496 views

Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product ...
4
votes
1answer
84 views

Hua Luogeng's definition of automorphism group for Hermitian symmetric space

I'm trying to make sense of a definition appearing in Hua Luogeng's book "Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains". Consider the Hermitian symmetric space ...
4
votes
1answer
219 views

Triviality of a fiber bundle

Is the principal fiber bundle $GL^+(6,\mathbb R)$ over $GL^+(6,\mathbb R)/SL(3,\mathbb C)$ trivial ?
2
votes
0answers
104 views

Intermediate quotient for a Hermitian Symmetric Spaces of $Sp(n)$

We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space: $$ SU(n+1)/U(n) \simeq {\mathbb CP}^{n}. $$ We can split this process into two ...
3
votes
0answers
61 views

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 ...
2
votes
0answers
57 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 ...
2
votes
0answers
39 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 ...
3
votes
1answer
148 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 ...
2
votes
1answer
100 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 ...
4
votes
1answer
147 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 ...
3
votes
1answer
251 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 ...
2
votes
2answers
135 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 ...
2
votes
1answer
109 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) = \{\, ...
2
votes
0answers
133 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 ...
5
votes
0answers
354 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 ...
2
votes
1answer
261 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 ...
5
votes
1answer
101 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 ...
5
votes
1answer
280 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. ...
1
vote
0answers
97 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 ...
4
votes
2answers
141 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 $-i$, $p+q=g$. ...
3
votes
0answers
61 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 ...
2
votes
1answer
268 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 ...
10
votes
1answer
390 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 ...
3
votes
0answers
157 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 ...
12
votes
0answers
549 views

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 ...
3
votes
1answer
142 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 ...
2
votes
1answer
156 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 ...
6
votes
2answers
207 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 ...
11
votes
2answers
340 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 ...
7
votes
3answers
275 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 ...
5
votes
1answer
252 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 ...
2
votes
0answers
83 views

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 ...
0
votes
0answers
106 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: ...
3
votes
2answers
450 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 ...
13
votes
1answer
385 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 ...
1
vote
2answers
281 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 ...
4
votes
1answer
362 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 ...
2
votes
1answer
143 views

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 ...
6
votes
2answers
281 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 ...
1
vote
0answers
229 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 ...
10
votes
0answers
473 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 ...
2
votes
1answer
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 ...
12
votes
2answers
2k views

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. ...
3
votes
3answers
637 views

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 ...
6
votes
1answer
457 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 ...
4
votes
1answer
378 views

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 ...
2
votes
2answers
879 views

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. ...
21
votes
2answers
1k views

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 ...
4
votes
0answers
209 views

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 ...
9
votes
3answers
2k views

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 ...