# 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 , ).
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 ).
A homogeneous space of an algebraic group $G$ is always a smooth quasi-projective variety (see ). 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 ). 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 , ).