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0
votes
0answers
47 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 ...
5
votes
1answer
73 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
218 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
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0answers
85 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
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2answers
108 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 $-i$, $p+q=g$. ...
1
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0answers
51 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 ...
1
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0answers
163 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
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1answer
319 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 ...
11
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0answers
426 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 ...
4
votes
1answer
119 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
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1answer
117 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
1answer
153 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 ...
10
votes
2answers
301 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
240 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
237 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
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0answers
75 views

Squared displacement function on Lorentzian manifolds

Hi, Let $\varphi$ be an isometry of a simply connected pseudo Riemannian manifold $M$. The squared displacement function of $\varphi$ is $d^2_{\varphi}(p):=d^2(\varphi (p),p)$, $p\in M$, where $d$ is ...
2
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0answers
79 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
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0answers
84 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: ...
0
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0answers
106 views

Some question on the paper “Ricci-flat metrics on the complexification of a compact rank one symmetric space”

I am reading the paper "Ricci-flat metrics on the complexification of a compact rank one symmetric space" by Stenzel (here is the ...
2
votes
2answers
315 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
333 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
242 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
346 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
137 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
272 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
212 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 ...
8
votes
0answers
395 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
244 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 ...
10
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
547 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 ...
5
votes
1answer
411 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
336 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 ...
1
vote
2answers
751 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. ...
20
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
198 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 ...
6
votes
1answer
571 views

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 ...
5
votes
3answers
537 views

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 ...
10
votes
4answers
944 views

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: ...
6
votes
1answer
445 views

Generalizing cosine rule to symmetric spaces

The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both ...
19
votes
2answers
1k views

“isotropic” subspaces of a simple Lie algebra

Let $\bf g$ be a finite-dimensional real simple Lie algebra of compact type and let $\left<-,-\right>$ denote the positive-definite inner product induced from the negative of the Killing form. ...