9
votes
Accepted
Relationship between fans and root data
(1) A (connected) reductive group $G$ over an algebraically closed field $k$ is described by a combinatorial object called the based root datum ${\rm BRD}(G)$.
(2) A spherical homogeneous space $Y=...
- 14.1k
8
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Accepted
Is the complement of the open $B$-orbit in a spherical variety cut out by one equation?
The answer is yes, even if $X$ is not normal. It even works for any connected solvable group acting on an affine variety with an open orbit.
To see this let $Y_1,\ldots,Y_r$ be the irreducible ...
- 15.5k
7
votes
Accepted
Is there a Chevalley map for spherical varieties?
Edit: The answer to question 1 is yes if $G/H$ is a symmetric variety as the OP pointed out.
For arbitrary spherical varieties the answer is no in general. If my memory serves me right, the spherical ...
- 15.5k
7
votes
Spherical roots, restricted roots, and the dual group of a symmetric variety
There is a small survey published in an Oberwolfach report by Bart Van Steirteghem comparing the different normalizations of spherical roots in more detail. The standard normalization is obtained by ...
- 15.5k
7
votes
Accepted
Uniqueness of the wonderful compactification of a semi-simple group
I assume you mean the variety $G$ considered as a $G \times G$ variety via the action $(g,h) \cdot x = gxh^{-1}$, which is the standard interpretation in the literature. The variety $G$ is spherical ...
- 1,780
7
votes
Accepted
Spherical and Wonderful varieties
The only groups which act on only one wonderful variety are tori (with $X$ being a point). All other admit at least $X=G/B$ and $X=G/G$.
If one fixes the open $G$-orbit then there is at most one ...
- 15.5k
7
votes
Accepted
Number of boundary divisors and colors of a Spherical variety
There are relations coming from computing Picard groups in different ways. For simplicity let $G$ be semisimple. Let $x\in X$ be in the open $B$-orbit. Then $Bx=B/B_x$ and $Gx=G/G_x$ are open in $X$.
...
- 15.5k
6
votes
Accepted
For a spherical pair $(G, H)$, which $G$-representations appear in $k[G/H]$?
Let $X_0=B/B_{x_0}=Bx_0$ be the open $B$-orbit in $X=G/H$. Then every character of $B$ which is trivial on $B_{x_0}$ yields a $B$-semiinvariant $f_\lambda$ on $X_0$. In general, it does not extend to ...
- 15.5k
5
votes
Accepted
Spherical roots, restricted roots, and the dual group of a symmetric variety
You might already have found an answer, but hopefully this helps.
The way I think about this is that there are always (at least) three normalizations of the roots to a spherical variety $X$. Some ...
- 2,516
5
votes
Accepted
Subvarieties of Lagrangian Grassmannians
Note first that, if $L$ is a Lagrangian contained in $H$ then $L^\perp = L$ contains $H^\perp$. So $X_n$ is non-empty only when $H$ is co-isotropic for your symplectic form.
When $H$ is co-isotropic, ...
- 1,908
5
votes
Accepted
Gelfand pair, weakly symmetric pair, and spherical pair
I like this question! I originally thought it was much less subtle, and so posted some ill informed guesses in the comments.
Although the notions of "weakly symmetric" and "Gelfand ...
- 12.9k
4
votes
Relationship between fans and root data
Not an answer, but: you can construct a fan from a root system. Let $R$ be a root system in an Euclidean space, and let $\Lambda_R$ be the root lattice with dual lattice $\Lambda_R^\vee$. The fan $\...
- 5,760
4
votes
Horospherical type of a spherical variety
As @LSpice pointed out, the group $S$ is not characterized by $N_G(S)=P^-$. Instead it is the normal subgroup of $P^-$ such that $P^-/S=T/T'$. Note also that $T'$ is in general not a subtorus of $T$ ...
- 15.5k
4
votes
Representations with finitely many nilpotent orbits
This is a sequence of remarks and a partial answer the question. I won't accept it at this point in case someone can say something more general/ intelligent.
Thanks to @Nandor for pointing out the ...
- 2,516
3
votes
Extending $p$-adic smooth and locally constant functions
Since $H^1(\mathbb Q_p, H)$ is finite and each fibre of $(G\cdot v)(\mathbb Q_p) \to H^1(\mathbb Q_p, H)$ is closed, we have that the fibre $G(\mathbb Q_p)\cdot v$ over the trivial cohomology class is ...
- 12.9k
3
votes
Accepted
Connected components of a spherical subgroup from spherical data?
Let $H^\circ\subseteq H$ be the connected component of unity and let $M^\circ$ be weight lattice of $X^\circ:=G/H^\circ$. Since $\pi_0(H)=H/H^\circ$ acts as automorphisms on $X^\circ$ it follows that $...
- 15.5k
3
votes
Accepted
Is any spherical subgroup conjugate to a subgroup defined over a smaller algebraically closed field?
The morphism $G/H\to\textrm{Spec}\,k$ is defined and flat over over some finitely generated $k_0$-subalgebra of $k$. Thus, the question amounts to whether a flat family of homomogeneous spherical $k_0$...
- 15.5k
3
votes
Accepted
Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$
The answer to both questions is "yes" as was communicated to me by Losev.
The main reason is that already $\varsigma$ is almost injective. More precisely, the sets $\varsigma(D)$ are always non-empty ...
- 15.5k
3
votes
Accepted
Morphisms of the spherical data of spherical homogeneous spaces
The situation is more complicated. First of all $\lambda_{\mathcal D}$ does not exist, since there may be colors in $X_1$ which map dominantly to $X_2$. The best way to describe morphisms $X_1\to X_2$ ...
- 15.5k
3
votes
points with small U stabilizer on a spherical variety
This set is a special case of regular semisimple elements on spherical varieties.
In particular it is open and dense.
- 2,639
2
votes
Finiteness of $H_1 \backslash G / H_2$ and the geometry of the orbits
This doesn't answer all your questions, but it seems relevant enough to post.
In The orbits of affine symmetric spaces under the action of minimal parabolic subgroups by T. Matsuki(1977), the double ...
- 8,529
2
votes
Accepted
Bialynicki-Birula decompositions and fixed points
I think all of these should be easy enough to resolve. First note that (1) is a triviality from your assumption that $G$ has finitely many orbits on $X$, because a maximal torus of $G$ can only have ...
- 134
2
votes
Accepted
Element of Weyl chamber contracting $\mathbb{A}^n_k$ to a point
The answer to your question, as stated, is No. Take $G={\rm SL}_2$, $P=B$, $T'=T$.
Note that you do not specify the isomorphism $T'\to {\Bbb G}_m^n$. Let us take the following isomorphism:
$$T'\to {\...
- 14.1k
2
votes
Accepted
An example of handle decomposition on modified $S^5$
I didn't really follow the details of your description, but here is a handle decomposition along the lines you suggest. Start with $((S^1 \times S^3)_a \coprod (S^1 \times S^3)_b) \times I$ where $I= [...
- 19.4k
1
vote
Subvarieties of Lagrangian Grassmannians
Co-isotropic subspaces of dimension $n+k$ are in bijection with isotropic subspaces of dimension $n-k$.
The variety of isotropic subspaces of dimension $n-k$ of a symplectic vector space of dimension $...
- 8,998
Only top scored, non community-wiki answers of a minimum length are eligible