6 - Homogeneous spaces and quotients

Summary

Let G be a linear algebraic group, X an algebraic variety. An action of G on X is a morphism G × X → X, written (g, x) ↦ gx, such that

1. (i) g(hx) = (gh)x for all g, h ∈ G and x ∈ X,

2. (ii) ex = x for all x ∈ X.

The variety X, equipped with an action of G as above, is called a G-variety or G-space. If X and Y are G-spaces, a morphism ϕ : X → Y is a G-morphism if ϕ(gx) = gϕ(x) for all g ∈ G and x ∈ X.

Let X be a G-space, x ∈ X. The isotropy group of x in G is the group G_x = {g ∈ G : gx = x}, the subgroup of G that fixes x. It is a closed subgroup of G, because it is the inverse image of {x} under the morphism g ↦ gx of G into X. The orbit of x is Gx = {gx : g ∈ G}, which is the image of G under the same morphism g ↦ gx, hence by (1.12) contains a non-empty open subset U of Gx. Since Gx is G-stable, it follows that the translates gU of U cover Gx, and hence that Gx is open in its closure Gx. This proves the first part of

(6.1) (i) Each orbit of G in X is a locally closed subvariety of X.