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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 asked me where 'spherical' came from and I had no idea. I asked a few more knowledgeable people and they also didn't know. So now I ask the same question here.

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I always imagined it traced back to $G/H = SO(n) / SO(n-1)$ describing the $n-1$-dimensional sphere as a homogeneous space. Here I think that a Borel $B_C$ acts with an open orbit on $G_C/H_C$, once one complexifies (hence the subscript $C$'s). –  Marty Jan 25 '11 at 20:55

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up vote 11 down vote accepted

Marty is definitely correct about the origin of the terminology in the study of homogeneous spaces $G/H$ of reductive Lie groups. Due to the special example involving a quotient of special orthogonal groups the handy term *spherical" got attached to the subgroup $H$ or the homogeneous space. In turn, this was carried over to the situation of algebraic groups and similar quotients in the algebraically closed case, where spherical varieties could be characterized as those homogeneous spaces admitting a dense orbit under a Borel subgroup. For instance, look at the introduction of an influential paper by Brion-Luna-Vust which appeared in Inventiones 84 (1986) with the title Espaces homogènes sphériques. While the label spherical variety is convenient, it loses its literal sense in this generality.

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Confirming Jim Humphreys' and Marty's answers, page 17 of (the English translation of) [Vinberg, È. B., Commutative homogeneous spaces and co-isotropic symplectic actions, MR1845642] contains:

In the simplest case of the two-dimensional sphere $S^2= SO(3)/SO(2)$ this result [the fact that the $SO(3)$-module $\mathbb{C}[S^2]$ is multiplicity-free] was known much earlier thanks to the Laplace spherical functions used in mathematical physics. This is the origin of the term “spherical homogeneous spaces”.

Note that if $X$ is an affine $G$-variety (with $G$ a complex reductive group), then $X$ has a dense orbit for a Borel subgroup $B \subset G$ if and only if $\mathbb{C}[X]$ is a multiplicity-free $G$-module.

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