# Questions tagged [spherical-varieties]

The spherical-varieties tag has no usage guidance.

38
questions

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### Extending $p$-adic smooth and locally constant functions

Let $G$ be $p$-adic group and let $G \rightarrow GL(V)$ be a representation. For example, $V$ is a quadratic $\mathbb{Q}_p$-space and $G$ is the associated orthogonal group.
Take a point $v \in V$, ...

1
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0
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111
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### Action of maximal compact subgroup on spherical homogeneous space $\DeclareMathOperator\S{SL}\S_2 \times \S_2 \times \S_2 / {\S_2}$

$\DeclareMathOperator\SL{SL}$Allow me to start my question by fixing some notations:
Let $X$ be the spherical homogeneous space $G / H$, where $G = \SL(2, \mathbb{C}) \times \SL(2, \mathbb{C}) \times \...

7
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0
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134
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### Quasisplit forms of wonderful varieties

I will assume that $k$ is a characteristic $0$ non-archimedean field. A classical result of Tits [T] states that a quasisplit connected reductive group $G$ over $k$ is classified up to strict isogeny ...

3
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1
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### Connected components of a spherical subgroup from spherical data?

This question is in a similar spirit to this one by Mikhail Borovoi.
Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
Losev proved that the spherical $...

4
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0
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110
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### Reference Request: Classification of spherical varieties by "Weyl group invariant fans"

Apologies in advance for the vague question.
Let $X$ be a spherical variety with the action of some reductive group $G$. I have been told in conversation several times that such spherical varieties ...

7
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0
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255
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### Status of the Luna's conjecture

In the famous IHES paper <Variétés Sphériques de Type A> of D. Luna, he proposed a conjecture asserting that wonderful varieties of an adjoint semisimple group $G$ are bijective to spherical ...

8
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2
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390
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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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0
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73
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### The local structure theorem for spherical varieties under quasi-split group action

I want to understand a simplified version of the general $k$-local structure theorem proved in the paper "Reductive group actions":
For $k$ a characteristic zero algebraically closed field, $...

3
votes

0
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244
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### When a stack quotient coincides with GIT quotient?

Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive.
Question: Is it true that when $G/H$ is open in its affine ...

1
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1
answer

179
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### An example of handle decomposition on modified $S^5$

I would like to give the following object, $M=S^5 \setminus \sqcup_{2 \text{ copies}} \text{int}(S^1\times D^4)$, a handle decomposition. It is then to be attached to another manifold. along the two ...

6
votes

1
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373
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### Is there a Chevalley map for spherical varieties?

If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $...

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0
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92
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### Why the trilinear GL_2 model is spherical?

Consider the homogeneous space $X:=GL_2\times GL_2\times GL_2/ H$ where $H=GL_2$ is diagonally embedded into $GL_2\times GL_2\times GL_2$. My question is why $X$ is spherical (i.e., there is a Borel ...

4
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2
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448
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### Subvarieties of Lagrangian Grassmannians

Let $LG(n,2n)$ be the Lagrangian Grassmannian parametrizing Lagrangian subspaces (so of dimension $n$) of $\mathbb{C}^{2n}$. Then $LG(n,2n)\subset G(n,2n)$, where $G(n,2n)$ is the Grassmannian of ...

3
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131
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### Localizations of smooth spherical varieties at simple roots

Setup
Let $G$ be a (connected) reductive group over an algebraically closed field $k$, and fix a Borel subgroup
$B \subset G$ and a maximal torus $T \subset B$. Let $\lambda: \mathbb{G}_m \to T$ be a ...

7
votes

1
answer

331
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### Gelfand pair, weakly symmetric pair, and spherical pair

I am a bit confused with the relations among Gelfand pairs, weakly symmetric pairs, and spherical pairs defined in the book "Harmonic analysis on commutative spaces" written by professor ...

6
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2
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### 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 ...

10
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1
answer

788
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### Bialynicki-Birula decompositions and fixed points

I was reading Luna's paper Toute variété magnifique est sphérique and stumbled on a few facts about Bialynicki-Birula decompositions and fixed points that I don't understand.
Here is the setup. Let $...

5
votes

1
answer

134
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### Element of Weyl chamber contracting $\mathbb{A}^n_k$ to a point

Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0. Fix a Borel subgroup $B$ and a maximal torus $T \subset B$. Let $P \subset G$ be a parabolic subgroup ...

12
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1
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301
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### For a spherical pair $(G, H)$, which $G$-representations appear in $k[G/H]$?

Let $G$ be a reductive algebraic group (over some alg. closed field $k$ of char 0), and $H$ a subgroup such that $(G, H)$ is spherical (i.e., the Borel $B$ of $G$ has an open orbit on $G/H$). Then $k[...

11
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1
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395
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### Finiteness of $H_1 \backslash G / H_2$ and the geometry of the orbits

Let $G$ be a connected reductive group over an algebraically closed field $k$. By the Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $...

1
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0
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116
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### Example of a spherical homogeneous space $G/H$ with a pairs of colors and with the center of $G$ not contained in $H$?

Let $G$ be a simply connected simple algebraic group over $\mathbb C$,
$B\subset G$ a Borel subgroup, and $T\subset B$ a maximal torus.
Let $\mathcal{S}=\mathcal{S}(G,T,B)$ denote the set of simple ...

2
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1
answer

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### Partial derivatives of spherical harmonics

Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?

4
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0
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### Compactifications of reductive groups via representation theory

Let $G$ be a reductive group, $\Lambda$ a weight lattice, $\Lambda^{+}$ the monoid of dominating weights, $\omega_1,\dots,\omega_r\in \Lambda^{+}$ the fundamental weights and $\{\alpha_1,\dots, \...

5
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1
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518
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### Uniqueness of the wonderful compactification of a semi-simple group

Let $G$ be a semi-simple group over an algebraically closed field of characteristic zero. In which cases there is a unique wonderful compactification of $G$ (modulo isomorphism)?
For instance, is the ...

6
votes

1
answer

676
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### Spherical and Wonderful varieties

A spherical variety is a normal variety $X$ together with an action of a connected reductive affine algebraic group $G$, a Borel subgroup $B\subset G$, and a base point $x_0\in X$ such that the $B$-...

4
votes

1
answer

192
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### Number of boundary divisors and colors of a Spherical variety

Let $X$ be a Spherical variety for a reductive group $G$ with a Borel subgroup $B$. A boundary divisor of $X$ is a $G$-invariant divisor and a color of $X$ is a $B$-invariant divisor which is no $G$-...

4
votes

1
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149
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### Is the complement of the open $B$-orbit in a spherical variety cut out by one equation?

Let $X$ be an affine spherical variety for some reductive algebraic group $G$. Let $X^0$ be the open orbit in $X$ under a fixed Borel subgroup $B \subseteq G$. Does there exists a function $f$ on $X$ ...

3
votes

1
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168
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### Is any spherical subgroup conjugate to a subgroup defined over a smaller algebraically closed field?

Let $G_0$ be a connected semisimple algebraic group defined over an algebraically closed field $k_0$. Let $k\supset k_0$ be a larger algebraically closed field.
We write $G=G_0\times_{k_0} k$ for the ...

4
votes

1
answer

217
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### Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$

Let $G$ be a semisimple group over $\mathbb C$, and $X=G/H$ be a spherical homogeneous space of $G$.
Let $T\subset B\subset G$ be a maximal torus and a Borel subgroup.
Let $S=S(G,T,B)$ denote the ...

5
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0
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163
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### The group of automorphisms a pair $(G,X)$ where $X$ is a spherical homogeneous space of $G$

I wish to "compute" ${{\rm Aut}}(G,X)$ for a spherical homogeneous space $X$ of $G$ in terms of the spherical datum of $X$.
First let $G$ be any algebraic group over $\mathbb C$, and let $X$ ...

1
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1
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148
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### Morphisms of the spherical data of spherical homogeneous spaces

Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a spherical homogeneous space,
then $X$ defines a spherical datum (Luna datum) $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\...

3
votes

1
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307
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### The group of $G$-automorphisms of a spherical variety from the spherical datum?

Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
By Losev's theorem, the spherical $G$-variety $X$ is uniquely determined by its spherical datum, see ...

4
votes

1
answer

213
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### A quotient group of a self-normalizing spherical subgroup

Let $G$ be simply connected, simple algebraic group over $\mathbb{C}$.
Let $H\subset G$ be a self-normalizing spherical subgroup of $G$,
not necessarily connected or reductive.
Here "self-...

17
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2
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1k
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### Is the wonderful compactification of a spherical homogeneous variety always projective?

Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}_G(H)=H$. Then by results of Brion and Pauer
...

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0
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273
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### GKZ decomposition for spherical varieties

If $X$ is a complete toric variety the GKZ decomposition of the effective cone $Eff(X)$ of $X$ corresponds to its Mori Chamber Decomposition, and therefore it encodes the birational geometry of $X$.
...

8
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### 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 ...

4
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1
answer

177
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### points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of $...

23
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3
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4k
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### Why are they called Spherical Varieties?

My understanding is if you have a homogeneous space $X = G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then you call $X$ spherical.
Someone ...