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In the following, I will fix $k$ a characteristic zero algebraically closed field, and $G$ a connected reductive group over $k$, $B$ a Borel subgroup of $G$, $T\subseteq B$ a maximal torus, $X$ a $G$-...
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Let $G$ be a reductive group over $\mathbb{C}$ and let $V$ be a finite dimensional representation of $G$. We can define the ``nilpotent cone'' of $V$ as $$\mathcal{N}(V):=\{ v\in V\;: \; 0\in\overline{...
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Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ ...
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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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$\DeclareMathOperator\SL{SL}$Allow me to start my question by fixing some notations: Let $X$ be the spherical homogeneous space $G / H$, where $G = \SL(2, \mathbb{C}) \times \SL(2, \mathbb{C}) \times \...
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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 ...
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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 $...
SLeslie's user avatar
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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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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 systems ...
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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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I want to understand a simplified version of the general $k$-local structure theorem proved in the paper "Reductive group actions": For $k$ a characteristic zero algebraically closed field, $...
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Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive. Question: Is it true that when $G/H$ is open in its affine ...
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I would like to give the following object, $M=S^5 \setminus \sqcup_{2 \text{ copies}} \text{int}(S^1\times D^4)$, a handle decomposition. It is then to be attached to another manifold. along the two ...
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If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $...
G. Gallego's user avatar
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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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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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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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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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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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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 $...
Michael Christianson's user avatar
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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 ...
Michael Christianson's user avatar
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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[...
David Loeffler's user avatar
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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 $...
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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 ...
Mikhail Borovoi's user avatar
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Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?
user124297's user avatar
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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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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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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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4 votes
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191 views

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 $X$ ...
Anonymous's user avatar
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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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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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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 $X$ ...
Mikhail Borovoi's user avatar
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1 vote
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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,\...
Mikhail Borovoi's user avatar
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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 ...
Mikhail Borovoi's user avatar
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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-...
Mikhail Borovoi's user avatar
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17 votes
2 answers
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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 ...
Mikhail Borovoi's user avatar
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1 vote
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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 $X$. ...
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8 votes
2 answers
436 views

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 ...
Rami's user avatar
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4 votes
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216 views

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 $...
Rami's user avatar
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23 votes
3 answers
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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 ...
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