Homogeneous space of an algebraic group

An algebraic variety $ M $ together with a regular transitive action of an algebraic group $ G $ given on it. If $ x \in M $ , then the isotropy group $ G _{x} $ is closed in $ G $ . Conversely, if $ H $ is a closed subgroup of an algebraic group $ G $ , then the set of left cosets $ G /H $ has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group $ G $ , where the natural mapping $ \pi : \ G \rightarrow G / H $ is regular, separable and has the following universal property: For any morphism $ \phi : \ G \rightarrow X $ constant on cosets, there is a morphism $ \psi : \ G / H \rightarrow X $ such that $ \psi \pi = \phi $ . If $ M $ is any homogeneous space of the algebraic group $ G $ and $ H = G _{x} $ for some $ x \in M $ , then the natural bijection $ \psi : \ G / H \rightarrow M $ is regular, and if the ground field $ K $ has characteristic 0, then $ \psi $ is biregular (see [1], [3]).

Suppose that the connected group $ G $ , the homogeneous space $ M $ and the action of $ G $ on $ M $ are defined over some subfield $ k \subset K $ . Then the group of $ k $ - rational points $ G (k) $ takes $ M (k) $ into itself and $ G (k) _{x} = G _{x} (k) $ for $ x \in M (k) $ . If $ k $ is finite, then $ M (k) \neq \emptyset $ , and if moreover the isotropy group $ G _{x} $ is connected, then $ G (k) $ acts transitively on $ M (k) $ . In the general case, the study of the $ k $ - rational points in $ M $ reduces to problems in the theory of Galois cohomology (see [2]).

A homogeneous space of an algebraic group $ G $ is always a smooth quasi-projective variety (see [5]). If $ G $ is an affine algebraic group, then the variety $ G / H $ is projective if and only if $ H $ is a parabolic subgroup in $ G $ ( see [3]). If $ G $ is reductive, then the variety $ G / H $ is affine if and only if the subgroup $ H $ is reductive (see Matsushima criterion). A description is also known of the closed subgroups $ H $ of a linear algebraic group $ G $ over an algebraically closed field of characteristic 0 for which $ G / H $ is quasi-affine (see [4], [6]).

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303
[3]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
[4]	A.A. Sukhanov, "A description of observable subgroups of linear algebraic groups" Math. USSR-Sb. , 68 (Forthcoming) Mat. Sb. , 137 : 1 (1988) pp. 90–102
[5]	W. Chow, "On the projective embedding of homogeneous varieties" , Algebraic topology; symposium in honour of S. Lefschetz , Princeton Univ. Press (1957) pp. 122–128 MR0084851 Zbl 0091.33302
[6]	G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) MR0620024 Zbl 0589.20025