Homogeneous space of an algebraic group

An algebraic variety together with a regular transitive action of an algebraic group given on it. If , then the isotropy group is closed in . Conversely, if is a closed subgroup of an algebraic group , then the set of left cosets has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group , where the natural mapping is regular, separable and has the following universal property: For any morphism constant on cosets, there is a morphism such that . If is any homogeneous space of the algebraic group and for some , then the natural bijection is regular, and if the ground field has characteristic 0, then is biregular (see [1], [3]).

Suppose that the connected group , the homogeneous space and the action of on are defined over some subfield . Then the group of -rational points takes into itself and for . If is finite, then , and if moreover the isotropy group is connected, then acts transitively on . In the general case, the study of the -rational points in reduces to problems in the theory of Galois cohomology (see [2]).

A homogeneous space of an algebraic group is always a smooth quasi-projective variety (see [5]). If is an affine algebraic group, then the variety is projective if and only if is a parabolic subgroup in (see [3]). If is reductive, then the variety is affine if and only if the subgroup is reductive (see Matsushima criterion). A description is also known of the closed subgroups of a linear algebraic group over an algebraically closed field of characteristic 0 for which 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) MR0181643 Zbl 0143.05901 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