Equivariant Embeddings of Algebraic Groups

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We conclude by describing G-equivariant morphisms between affine G-embeddings in terms of this classification. In particular, if X 1 → X 2 is a morphism of G-embeddings and x i ∈ X i are base points such that x 1 ↦→ x 2 , then Γ(X 1 , x 1 ) ⊆ Γ(X 2 , x 2 ) and conversely. We characterize biequivariant affine G-embeddings in terms of their cones and construct a canonical “biequivariant resolution” X G → X of any given affine G-embedding X, where X G is a (G × G)-equivariant affine G-embedding satisfying a universal property. 10
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. 11
Page 1 and 2: c○ Copyright by David Charles Mur
Page 3 and 4: Abstract We classify embeddings of
Page 5 and 6: Acknowledgements This thesis would
Page 7 and 8: Chapter 1 Introduction A basic prob
Page 9 and 10: x i ≠ 0} for i = 0, 1, 2, the one
Page 11 and 12: complement of U in G/H. Remark 1 ([
Page 13 and 14: G. The wonderful compactification o
Page 15: X is to use one-parameter subgroups
Page 19 and 20: Proposition 3 ([28], Proposition I.
Page 21 and 22: and every toric variety for T arise
Page 23 and 24: To see this, suppose D is a T -stab
Page 25 and 26: Conversely, every subdivision of C
Page 27 and 28: 0 : 0 : 1 : 1], [0 : 0 : 0 : 1 : 1]
Page 29 and 30: Mumford’s embedding is constructe
Page 31 and 32: Moreover, Ind(X) is a G-embedding,
Page 33 and 34: the T -action with respect to the c
Page 35 and 36: from X with a line bundle from the
Page 37 and 38: G/B × G/B − corresponding to the
Page 39 and 40: since G (and hence G × G) is semi-
Page 41 and 42: 1. There is a finite-dimensional G-
Page 43 and 44: the following (obviously) commutati
Page 45 and 46: Now, to prove the claim, by Lemma 7
Page 47 and 48: ational polyhedral cone in X ∗ (T
Page 49 and 50: ⋃ P ⊃T P • (σ ∩ ∆ P (T )
Page 51 and 52: Section 3.2.3, is a uniqueness theo
Page 53 and 54: Therefore, the classification of af
Page 55 and 56: For each k-point g of G, we have ma
Page 57 and 58: is over a finite set of integers J.
Page 59 and 60: i ≥ 0, 〈χ ij , γ (0,1) 〉 =
Page 61 and 62: Lemma 12. If γ 1 , γ 2 ∈ X ∗
Page 63 and 64: an integrally closed left-invariant
Page 65 and 66: of the subalgebras A Γ(X,x0 ) ⊂
Page 67 and 68:
on varieties discussed in this sect
Page 69 and 70:
Proof. Let Γ be a strongly convex
Page 71 and 72:
that Γ(X, [I 2 , (1, 1)]) GL 2 = {
Page 73 and 74:
References [1] A. Bialynicki-Birula
Page 75 and 76:
[24] B. Iversen, Cohomology of Shea
Page 77:
Vita David Charles Murphy was born
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Equivariant Embeddings of Algebraic Groups

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