Equivariant Embeddings of Algebraic Groups

Definition 2. 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 an affine G-orbit,
∂X is a G-stable divisor of X unless Ω = X.
Any toric variety is a T -embedding, in our terminology. Similarly, the wonderful compactification
of an adjoint group G defined by De Concini and Procesi [7] is a G-embedding. All of these
examples have both a left and a right G-action. Yet our definition of a G-embedding allows the
consideration of G-varieties which only possess a left action of the group. For instance, if Z is a
toric variety for a maximal torus T of a reductive group G, then the variety X = G × T Z is a
normal G-variety for the action g · [h, z] = [gh, z], and the orbit G · [e G , e T ] ⊂ X is open and isomorphic
to G. However, we can’t define a right action of G on X since we can’t do so in a way that
preserves the equivalence relation (g, z) ∼ (gt −1 , tz) defined by the diagonal action of T on G × Z.
Thus, our definition of G-embeddings includes all of the biequivariant compactifications already
in the literature [3], [6], [7], [28], [46] and many more. In Chapter 2, we discuss these biequivariant
G-embeddings as well as other constructions of G-embeddings derived from toric varieties.
An important consequence of this discussion is the proof that non-isomorphic G-embeddings can
produce isomorphic toric varieties T , which is impossible in the (G × G)-equivariant case. This
is accomplished by our construction of a family of pairwise non-isomorphic SL n -embeddings in
Section 2.2.2 in which two or more of the induced toric varieties T are isomorphic whenever n ≥ 4.
Suppose X is a G-embedding. Identify G with the open orbit Ω in X by selecting a base point
x 0 ∈ Ω and considering the isomorphism G → Ω given by g ↦→ g · x 0 . The choice of base point
x 0 ∈ Ω, and hence the identification of G with Ω, is not canonical. If x ′ is any other point of Ω,
then it could serve just as well to identify G with Ω. For such x ′ ∈ Ω, there is a unique element
h ∈ G such that x ′ = h · x 0 . Thus any base point differs from a fixed one by a unique element of
G. The boundary is unchanged, but the manner in which G is identified with Ω changes the way
in which one-parameter subgroups of G can approach ∂X, as we will see below. We will want our
classification to be independent of this choice of base point x 0 , but for now our constructions will
rely upon it.
Suppose that X is an affine G-embedding with base point x 0 , so X is a normal affine G-variety
containing the open orbit Gx 0 isomorphic to G. Our primary method for classifying embeddings
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