# Questions tagged [spherical-varieties]

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28
questions

**6**

votes

**2**answers

276 views

### 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**

votes

**0**answers

77 views

### 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 ...

**6**

votes

**1**answer

113 views

### 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**

votes

**2**answers

295 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 ...

**10**

votes

**1**answer

353 views

### 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

101 views

### 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 ...

**10**

votes

**1**answer

233 views

### 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[...

**6**

votes

**0**answers

217 views

### Is there a scheme-theoretic description of spherical varieties?

I haven't found much expository literature, and in particular, I'd like some references on how spherical varieties could be viewed schematically, especially functorially (i.e. viewed from the functor ...

**11**

votes

**1**answer

368 views

### 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**

vote

**0**answers

100 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 ...

**2**

votes

**1**answer

2k 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?

**2**

votes

**0**answers

139 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

199 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

393 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

338 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

137 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

**1**answer

129 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

162 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

189 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

151 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 $X$ ...

**1**

vote

**1**answer

117 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

227 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

181 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-...

**16**

votes

**2**answers

993 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

226 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

334 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 ...

**4**

votes

**1**answer

156 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 $...

**22**

votes

**3**answers

3k 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 ...