Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Chapter 3<br />
Affine group embeddings<br />
Let G be a connected reductive algebraic group defined over an algebraically closed field k <strong>of</strong><br />
characteristic 0. By a G-variety, we mean a k-variety X together with a morphism σ : G × X → X,<br />
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,<br />
where e ∈ G denotes the identity element. We are interested in the following class <strong>of</strong> G-varieties.<br />
Definition 8. A G-embedding is a normal G-variety X that contains an open orbit Ω isomorphic<br />
to G. The closed subvariety ∂X = X − Ω is called the boundary <strong>of</strong> X. Since Ω is a G-orbit, ∂X<br />
is a G-stable divisor <strong>of</strong> X unless Ω = X. The irreducible components <strong>of</strong> ∂X are G-stable prime<br />
divisors <strong>of</strong> X.<br />
Before we begin our classification <strong>of</strong> affine G-embeddings in this chapter, we study the nature<br />
<strong>of</strong> an action <strong>of</strong> a reductive group G on an affine variety X, and, in particular, the induced actions<br />
<strong>of</strong> the one-parameter subgroups <strong>of</strong> G.<br />
3.1 Group actions on affine varieties<br />
In this section, we recall the results <strong>of</strong> [27] that, with their pro<strong>of</strong>s, will be exploited throughout the<br />
chapter. Some <strong>of</strong> this material also appears in [32].<br />
Let X be an affine variety with an action by a reductive group G, σ : G × X → X. This action<br />
corresponds to the coaction σ ◦ : k[X] → k[G] ⊗ k[X], which will be critical for the following.<br />
Lemma 7 ([27], Lemma 1.1). Let X be an affine G-variety and suppose that Y is a closed<br />
G-stable subvariety <strong>of</strong> X. Then:<br />
34