Equivariant Embeddings of Algebraic Groups

Chapter 3
Affine group embeddings
Let G be a connected reductive algebraic group defined over an algebraically closed field k of
characteristic 0. By a G-variety, we mean a k-variety X together with a morphism σ : G × X → X,
written (g, x) ↦→ g · x, satisfying g 1 · (g 2 · x) = (g 1 g 2 ) · x and e · x = x for all g 1 , g 2 ∈ G and all x ∈ X,
where e ∈ G denotes the identity element. We are interested in the following class of G-varieties.
Definition 8. A G-embedding is a normal G-variety X that contains an open orbit Ω isomorphic
to G. The closed subvariety ∂X = X − Ω is called the boundary of X. Since Ω is a G-orbit, ∂X
is a G-stable divisor of X unless Ω = X. The irreducible components of ∂X are G-stable prime
divisors of X.
Before we begin our classification of affine G-embeddings in this chapter, we study the nature
of an action of a reductive group G on an affine variety X, and, in particular, the induced actions
of the one-parameter subgroups of G.
3.1 Group actions on affine varieties
In this section, we recall the results of [27] that, with their proofs, will be exploited throughout the
chapter. Some of this material also appears in [32].
Let X be an affine variety with an action by a reductive group G, σ : G × X → X. This action
corresponds to the coaction σ ◦ : k[X] → k[G] ⊗ k[X], which will be critical for the following.
Lemma 7 ([27], Lemma 1.1). Let X be an affine G-variety and suppose that Y is a closed
G-stable subvariety of X. Then:
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