All Questions
Tagged with symmetric-spaces ag.algebraic-geometry
23 questions
12
votes
0
answers
246
views
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: ...
- 2,516
11
votes
4
answers
2k
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: ...
- 2,713
8
votes
2
answers
459
views
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 ...
- 553
8
votes
2
answers
386
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 ...
- 2,639
7
votes
0
answers
235
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 $\mu:M\times M\to M$ ...
- 1,077
6
votes
2
answers
430
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 ...
- 4,081
6
votes
1
answer
287
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 ...
- 7,902
5
votes
3
answers
786
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 ...
- 33.3k
5
votes
1
answer
260
views
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 ...
- 6,552
5
votes
1
answer
680
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
...
user5117
5
votes
1
answer
321
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 N_G(G^\...
- 3,968
4
votes
3
answers
264
views
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. ...
- 741
4
votes
0
answers
108
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 ...
- 2,221
3
votes
1
answer
227
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 ...
- 980
3
votes
0
answers
85
views
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}(\...
- 796
2
votes
2
answers
1k
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. ...
- 741
2
votes
1
answer
294
views
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 ...
- 21
2
votes
1
answer
176
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 ...
- 27.8k
2
votes
0
answers
170
views
Orbits under an algebraic group inside a Shimura variety
Let $(G,X)$ be a Shimura datum, $K$ a compact open subgroup of $G(\mathbb A_f)$. Let $H\subset G$ be a $\mathbb Q$-subgroup. Choose a point $x\in X$. What can be said about the orbit $H(\mathbb R)\...
- 796
2
votes
0
answers
118
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 ...
- 2,639
2
votes
0
answers
129
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 $\...
- 1,780
1
vote
0
answers
96
views
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 ...
- 4,386
0
votes
0
answers
75
views
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 $0$....
- 519