# What does a homogeneous space of a linear algebraic group know about the group?

Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers and $H\subset G$ is an algebraic subgroup. In general, we can write the algebraic variety $X$ as $X=G'/H'$ with $G'$ non-isomorphic to $G$. What can be said about $G'$ (and $H'$)?

Of course, we can always take any surjective homomorphism $G'\to G$, then $X$ is naturally a homogeneous space of $G'$. Conversely, if the action of $G$ on $X$ is not effective, we can take a normal subgroup $N$ of $G$, contained in $H$ (say, the kernel of the action) and set $G'=G/N,\ H'=H/N$, then $X=G'/H'$. I am interested in less trivial examples.

I would be happy to get a general answer, but will be also grateful for special cases, examples, comments, etc.

I am obliged to Günter Harder for this nice title of my question.

For the special case where $X$ is projective, and reducing (by the "trivial" examples of surjections and normal quotients) to $G$ a semisimple group of adjoint type, with $H=P$ a parabolic, one version of the answer is given by Demazure in "Automorphismes et déformations des variétés de Borel", Invent. Math., 1977. In a nutshell, the automorphism group of such an $X$ is the subgroup of $Aut(G)$ preserving the conjugacy class of $P$ --- unless the pair $(G,P)$ is one of several "exceptional" cases. (E.g., the $5$-dimensional quadric is both $G_2/P$ and $SO_7/P$.)
In other words, for the non-exceptional cases, you can recover $G$ as the connected component of the identity in $Aut(X)$.
• Two comments: 1) The underlying field need not be of characteristic 0 in Demazure's treatment (thanks to Kempf's vanishing theorem in char $p$). 2) This and some other important articles by Demazure can be downloaded from gdz.sub.uni-goettingen.de/; just type "Demazure" in the search box. Feb 24, 2011 at 13:56
To add perspective to Dave's answer, it seems that homogeneous spaces realized as quotients of a connected reductive group $G$ by parabolic subgroups are often natural and rich in structure besides having close connections with $G$. But I'm less confident about other types of homogeneous spaces. There are lots of arbitrary constructions, e.g., start with a product of connected 1-dimensional groups and factor out the product of one or more of them. At that extreme the homogeneous space has little connection with the given group.
Even though factoring out a parabolic subgroup (especially a maximal one) tends to produce a relatively "small" variety, its automorphism group is often close to $G$ (as seen in Demazure's paper). The only other type of maximal closed connected proper subgroup in $G$ is itself reductive, as follows from Borel-Tits theory in arbitrary characteristic: see for instance Section 30.4 of my Springer text GTM 21. But then you get an affine homogeneous space. In fact, R.W. Richardson's classic theorem in Bull. London Math. Soc. 9 (1977) shows that $G/H$ is affine iff the closed subgroup $H$ of the reductive group $G$ is itself reductive.