All Questions
16
questions
7
votes
0
answers
134
views
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 ...
- 121
3
votes
1
answer
124
views
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 $...
- 2,386
1
vote
0
answers
92
views
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 ...
- 1,111
6
votes
2
answers
410
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 ...
- 3,991
10
votes
1
answer
787
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
134
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 ...
11
votes
1
answer
395
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 $...
- 6,188
1
vote
0
answers
116
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 ...
- 13.8k
5
votes
1
answer
518
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 ...
user117617
4
votes
1
answer
149
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 $X$ ...
- 413
3
votes
1
answer
168
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 ...
- 13.8k
4
votes
1
answer
217
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 ...
- 13.8k
4
votes
1
answer
213
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-...
- 13.8k
17
votes
2
answers
1k
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
...
- 13.8k
8
votes
2
answers
376
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 ...
- 2,591