# When is an orbit spherical?

I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:

Let's assume we have an affine, reductive, algebraic group $G$ acting algebraically on a variety $X$, everything over an algebraically closed field of characteristic zero. Let $x\in X$ be some point with reductive stabilizer $H:=G_x$. Under what conditions on $x$ or $H$ is the orbit $G.x\cong G\newcommand{\qq}{/\hspace{-.8ex}/}\qq H$ a spherical variety? Let me briefly recall that a spherical variety is a homogeneous space $G\qq H$ satisfying one of the following, equivalent properties:

1. Any Borel subgroup $B\subseteq G$ has an open orbit in $G\qq H$.
2. Every equivariant completion of $G\qq H$ contains only finitely many orbits.
3. For every irreducible $G$-module $V$ and any character $\chi$ of $H$, $$\dim\left\{~v\in V \mid \forall h\in H: h.v = \chi(h)v ~\right\}\le 1.$$

I was hoping that this is well-known, but I cannot find any direct statements of that kind. Searching for the keywords "orbit" and "spherical" is quite fruitless because of property 1.

Edit: In the cases of interest to me, the orbit $G.x$ is affine.

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@Jesko: I think your affineness condition on the orbit $G/H$ is equivalent to your hypothesis that the underlying reduced scheme of the identity component $H^0$ is reductive. This equivalence is a theorem of Borel & Harish-Chandra in characteristic 0 (proved via topological methods over $\mathbf{C}$), Richardson in any characteristic (proved via Haboush's work on Mumford's conjecture in GIT), and finally proved by Borel in any characteristic via etale cohomology (adapting the argument with H-C). –  grp Sep 7 '12 at 11:59
@grp: Actually, Richardson's work is in characteristic 0 but based more on Grothendieck's algebraic geometry, whereas the work of Haboush was aimed at prime characteristic where Mumford's conjecture was then open. (A paper by Cline-Parshall-Scott extended the results on affine quotients to prime characteristic.) –  Jim Humphreys Sep 7 '12 at 22:01
@Jesko: Maybe an added tag algebraic-groups would be appropriate? –  Jim Humphreys Sep 7 '12 at 22:02
@Jim Humphreys: I think that what I wrote was accurate: in Theorem A of Richardson's paper "Affine coset spaces of reductive algebraic groups" in Bulletin of the LMS 9 (1977), pp. 38--41, he uses Haboush's theorem (Mumford's conjecture) to prove the result in arbitrary characteristic. He states explicitly early in the paper that his purpose is to record a proof valid in any characteristic. I'm not sure which papers of Richardson and C-P-S you have in mind, but please look at the Richardson paper I have just mentioned. –  grp Sep 8 '12 at 5:47
@grp: Sorry for the clumsy wording. Richardson's main point was to rewrite the analytic proof in algebraic language via Grothendieck; he could then incorporate char p as well, using Haboush's theorem (reductive implies geometrically reductive). That affirmed Nagata's earlier approach based on assuming Mumford's conjecture. But the question here is just about char 0. In their 1977 paper (Math. Ann. 230), CPS gave a broader treatment in char p, showing how induction functors come into the picture.. –  Jim Humphreys Sep 8 '12 at 11:33

This was an active problem 20 years ago. I don't know if it's completely resolved, but more can be said for affine spherical homogeneous spaces. (Also---in general, a spherical variety need not be homogeneous, so searching for "spherical homogeneous variety might yield better results.) Try Brion, "Classification des espaces homogènes sphériques" (Compositio, 1987) and more recent work of Knop et al, e.g., http://arxiv.org/abs/math/0505102 . The latter has convenient tables at the end.

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I added an edit noting that the cases of interest to me are affine orbits anyway, so this is great news. I will look into that right away, thanks a lot already. –  Jesko Hüttenhain Sep 7 '12 at 11:48

An extended comment: As Dave points out, Michel Brion has been active in the study of spherical varieties (and some generalizations) in the setting of reductive algebraic groups in characteristic 0. His work spans by now several decades, up to the present, and includes research papers in both French and English along with numerous surveys and lectures. His classification result in the affine case (i.e., $H$ affine) was already outlined in his note:

Classification des espaces homogenes spheriques. C. R. Acad. Sci. Paris S´er. I Math. 301 (1985), no. 18, 813–815.

A number of these papers are published in Comment. Math. Helv., while an influential joint paper with Luna and Vust can be found online at GDZ:

Espaces homogenes spheriques. Invent. Math. 84 (1986), no. 3, 617–632.

I'm not sure whether this extensive work, or that of Knop et al. linked by Dave, will have all the information you want, but the subject remains active.

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