Questions tagged [spherical-varieties]

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5
votes
2answers
243 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
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1answer
188 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
1answer
86 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 ...
9
votes
1answer
218 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[...
5
votes
0answers
206 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
1answer
363 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
0answers
98 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
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1answer
1k 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
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0answers
128 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
0answers
185 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
1answer
351 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
1answer
286 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
1answer
122 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
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0answers
157 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
1answer
123 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
1answer
160 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
1answer
181 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
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0answers
147 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
1answer
107 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
1answer
211 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
1answer
168 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" ...
16
votes
2answers
958 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
0answers
203 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
2answers
320 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
1answer
152 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
3answers
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 ...