# Questions tagged [spherical-varieties]

The spherical-varieties tag has no usage guidance.

**1**

vote

**0**answers

77 views

### 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 ...

**1**

vote

**2**answers

414 views

### 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?

**6**

votes

**0**answers

134 views

### Spherical varieties as GIT quotients

Let $X$ be a normal projective variety with finitely generated Cox ring. Consider its characteristic space $p:\widehat{X}\rightarrow X$. This means that there is a torus $T$ acting on $\overline{X}=...

**2**

votes

**0**answers

115 views

### On the classification of spherical varieties

Let $G$ be a connected reductive algebraic group, for instance take $G = SL_n$. Does there is a classification of the $\mathbb{Q}$-factorial normal projective varieties with given dimension and Picard ...

**8**

votes

**0**answers

162 views

### 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, \...

**6**

votes

**1**answer

256 views

### 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

208 views

### 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 $B$-...

**5**

votes

**1**answer

100 views

### 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

**0**answers

133 views

### Generalizations of Spherical varieties

A spherical variety is a normal projective variety $X$ with an action of a reductive group $G$ admitting a Borel subgroup $B\subseteq G$ such that $B$ has an orbit which is dense in $X$.
I wanted to ...

**4**

votes

**1**answer

109 views

### 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**answer

151 views

### 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 ...

**3**

votes

**1**answer

166 views

### 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**

votes

**0**answers

137 views

### 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 $X$ be any ...

**1**

vote

**1**answer

99 views

### 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,\...

**1**

vote

**1**answer

193 views

### 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

148 views

### 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-normalizing" ...

**15**

votes

**2**answers

900 views

### 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
...

**2**

votes

**0**answers

154 views

### 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**

votes

**2**answers

308 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 ...

**3**

votes

**1**answer

146 views

### 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 $...

**24**

votes

**3**answers

2k views

### 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 ...