Equivariant Embeddings of Algebraic Groups

Chapter 2
Toric varieties and group embeddings
Throughout this chapter, k is an algebraically closed field of characteristic zero, G will denote a
connected reductive algebraic group defined over k, and T will denote a maximal torus of G.
Suppose X is a G-embedding. That is, assume X is a normal G-variety which contains G as
an open subvariety in such a way that the action of G on X extends the left translation action
on G. Note that this is a generalization of the notion of toric varieties to more general algebraic
groups. The hope for such a theory was mentioned by Danilov in [12]. However, little work in this
generality appears in the literature, where preference is given to the class of embeddings that are
also spherical varieties for G × G.
Among the biequivariant embeddings of G, i.e., G-embeddings which have both a left and a right
action by G, the regular compactifications of G have been studied extensively [3], [2], [5], [46], [44]
and, more specifically, the wonderful compactification of the group [7], [8], [9], [41], [6]. In such
embeddings, the closure of a maximal torus is always a toric variety [46] and this toric variety
determines the compactification [5].
In this chapter, we investigate the relationship between toric varieties and embeddings of reductive
groups in a number of contexts. First, we will briefly review the theory of toric varieties,
borrowing heavily from [28], [12], [33], [17], and [14]. With this background in hand, we next consider
toric varieties in group embeddings. Finally, we conclude the chapter with several constructions of
group embeddings using toric varieties. Using one of these constructions, in Section 2.2.2 we prove
that the closure of a single maximal torus of G in a G-embedding is insufficient for determining the
embedding, in contrast to the result mentioned above for regular compactifications.
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