# Questions tagged [spherical-varieties]

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### Extending $p$-adic smooth and locally constant functions

Let $G$ be $p$-adic group and let $G \rightarrow GL(V)$ be a representation. For example, $V$ is a quadratic $\mathbb{Q}_p$-space and $G$ is the associated orthogonal group. Take a point $v \in V$, ...
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### Reference Request: Classification of spherical varieties by "Weyl group invariant fans"

Apologies in advance for the vague question. Let $X$ be a spherical variety with the action of some reductive group $G$. I have been told in conversation several times that such spherical varieties ...
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### Status of the Luna's conjecture

In the famous IHES paper <Variétés Sphériques de Type A> of D. Luna, he proposed a conjecture asserting that wonderful varieties of an adjoint semisimple group $G$ are bijective to spherical ...
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### Spherical roots, restricted roots, and the dual group of a symmetric variety

Let $k$ be an algebraically closed field of characteristic $0$ and $G$ a semisimple simply-connected group over $k$. Consider a symmetric variety of the form $X=G/H$, for $H=G^\theta$ the fixed point ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $B$-...
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### 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 $G$-...
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### 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 $X$ ...
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### 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 ...
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### 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 ...
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### 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 $X$ ...
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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) $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\... • 13.8k 3 votes 1 answer 307 views ### The group of$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$is uniquely determined by its spherical datum, see ... • 13.8k 4 votes 1 answer 213 views ### A quotient group of a self-normalizing spherical subgroup Let$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$. Then by results of Brion and Pauer ... • 13.8k 1 vote 0 answers 273 views ### GKZ decomposition for spherical varieties If$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$. ... 8 votes 2 answers 376 views ### regular semisimple elements on spherical varieties Let$(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$). What can ... • 2,591 4 votes 1 answer 177 views ### points with small U stabilizer on a spherical variety Let$(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$be the unipotent radical of$...
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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 ...