Equivariant Embeddings of Algebraic Groups

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g[Γ ∩ X ∗ (T )]g −1 = gσg −1 . Thus Γ = ⋃ g∈G gσg−1 is determined by the cone σ, so X is determined by its toric subvariety T = T σ . We note the similarity between our classification in the case of biequivariant affine G-embeddings and Brion’s classification of regular G-compactifications in Section 2.2.1. In [5], Brion classified regular compactifications of a group G by the W (T, G)-invariant fan associated to the closure of any maximal torus T of G in the compactification, demonstrating that regular compactifications of G are completely determined by any of the associated toric subvarieties. Likewise, Corollary 4 implies that if an affine G-embedding X is (G × G)-equivariant and T is a maximal torus of G, then the cone for T recovers X. Therefore, we may think of Proposition 10 and Corollary 4 as an affine version of Brion’s classification. Remark 5. Suppose X is a (G × G)-equivariant affine G-embedding and suppose that x 0 ∈ X is a base point. By Proposition 10 above, the set Γ(X, x 0 ) is stable under the conjugation action of G. However, by formula (3.7), we conclude that Γ(X, x 0 ) = hΓ(X, x 0 )h −1 = Γ(X, h · x 0 ) for all h ∈ G. Therefore, the strongly convex lattice cone associated to a biequivariant affine G-embedding is independent of the choice of base point. Hence, let X be a biequivariant affine G-embedding. By formula (3.14), if Ξ is a Kempf substate generating Γ(X, x) ∨ , then g ∗ Ξ is a Kempf substate that generates Γ(X, g · x) ∨ . Hence, if Γ is a G- stable strongly convex lattice cone whose monoid state is generated by the Kempf substate Ξ, then the Kempf state G ∗ Ξ : R ↦→ ⋃ g∈G g ∗Ξ(R) generates Γ ∨ as well and is G-invariant for conjugation. We use these observations to construct the biequivariant resolution of an affine G-embedding X. Suppose X is an affine G-embedding, x ∈ X is a base point, and Γ(X, x) is the corresponding strongly convex lattice cone. Then there is a unique maximal G-stable subset Γ G := ⋂ hΓh −1 (3.18) of Γ. In fact, h∈G Lemma 13. If Γ is a strongly convex lattice cone in X ∗ (G), then Γ G = ⋂ h∈G hΓh−1 is a strongly convex lattice cone. 62
Proof. Let Γ be a strongly convex lattice cone. Thus Γ ∨ is a monoid state, Γ = (Γ ∨ ) ∨ , and γ, γ −1 ∈ Γ if and only if γ is trivial. As Γ ∨ is a monoid state, let Ξ be a Kempf substate generating Γ ∨ . Define Γ G = ⋂ h∈G hΓh−1 as in (3.18). We claim that G ∗ Ξ : R ↦→ ⋃ g∈G g ∗Ξ(R) is a Kempf state and that it generates (Γ G ) ∨ , in which case Γ G is a lattice cone. First, G ∗ Ξ is a state, for each g ∗ Ξ is a state implies that G ∗ Ξ(R) is nonempty for all R and that if S ⊃ R, then G ∗ Ξ(R) = ⋃ g∈G g ∗Ξ(R) = ⋃ [ ⋃ ] g∈G resS R [g ∗Ξ(S)] = res S R g∈G g ∗Ξ(S) = res S R [G ∗Ξ(S)]. Next, G ∗ Ξ is bounded, for if h ∈ G, then h ∗ [G ∗ Ξ](R) = ⋃ g∈G h ∗[g ∗ Ξ(R)] = ⋃ g∈G (hg) ∗Ξ(R) = G ∗ Ξ(R), so ⋃ h∈G h ∗[G ∗ Ξ](R) = G ∗ Ξ(R) = ⋃ g∈G g ∗Ξ(R) is a finite subset of X ∗ (R) for each R, since Ξ is a bounded state. function µ(G ∗ Ξ) is defined by Finally, G ∗ Ξ is admissible, for its numerical µ(G ∗ Ξ, γ) = min 〈χ, γ〉 = min χ∈G ∗Ξ(γ) χ∈ ⋃ g ∗Ξ(γ) 〈χ, γ〉 = min min g∈G χ∈g ∗Ξ(γ) 〈χ, γ〉 = min µ(g ∗Ξ, γ) g∈G and each g ∗ Ξ is admissible. That is, if γ is a one-parameter subgroup of G and p ∈ P (γ) belongs to the parabolic subgroup associated to γ, then µ(G ∗ Ξ, p • γ) = min g∈G µ(g ∗ Ξ, p • γ) = min g∈G µ(g ∗ Ξ, γ) = µ(G ∗ Ξ, γ). Thus G ∗ Ξ is a Kempf state. Now we show that each G ∗ Ξ(R) generates (Γ G ) ∨ (R), for all tori R of G. Suppose χ ∈ (Γ G ) ∨ (R) for some torus R of G. Now (Γ G ) ∨ (R) = ( ⋂ g∈G gΓg−1 ) ∨ (R) = ∑ g∈G (gΓg−1 ) ∨ (R), so χ is a sum of elements from the monoids (gΓg −1 ) ∨ (R). As the sets g ∗ Ξ(R) generate the monoids (gΓg −1 ) ∨ (R), we may write χ = ∑ g∈G (∑ ξ∈g n ∗Ξ(R) ξξ) for non-negative integers n ξ , which is a finite sum since ∑ ∑ g∈G ξ∈g = ∑ ∗Ξ(R) ξ∈G and G ∗Ξ(R) ∗Ξ(R) is finite. Thus G ∗ Ξ(R) generates (Γ G ) ∨ (R) as a monoid for each R. Therefore, to prove that (Γ G ) ∨ is a monoid state, it suffices now to show that (Γ G ) ∨ is a state, for we have shown that G ∗ Ξ will be a suitable Kempf substate. Clearly each (Γ G ) ∨ (R) is nonempty, for it will contain at least the zero character. Assume R ⊂ S is an inclusion of tori of G. Then res S R [(ΓG ) ∨ (S)] ⊂ (Γ G ) ∨ (R) by definition of (Γ G ) ∨ as mentioned after formula (3.16). Now suppose that χ ∈ (Γ G ) ∨ (R). We may write χ = ∑ ξ∈G ∗Ξ(R) n ξξ for non-negative integers n ξ since (Γ G ) ∨ (R) is generated as a monoid by G ∗ Ξ(R). Yet G ∗ Ξ is a state, so each ξ ∈ G ∗ Ξ(R) is the restriction of some not-necessarily-unique ξ S ∈ G ∗ Ξ(S). Let χ S = ∑ ξ∈G ∗Ξ(R) n ξξ S ∈ (Γ G ) ∨ (S) and consider res S R (χS ) = ∑ ξ∈G ∗Ξ(R) n ξres S R (ξS ) = ∑ ξ∈G ∗Ξ(R) n ξξ = χ. Hence res S R [(ΓG ) ∨ (S)] = (Γ G ) ∨ (R), which proves that (Γ G ) ∨ is a state. Therefore, (Γ G ) ∨ is a monoid state and Γ G is a lattice cone in 63
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c○ Copyright by David Charles Mur
Page 3 and 4:
Abstract We classify embeddings of
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Acknowledgements This thesis would
Page 7 and 8:
Chapter 1 Introduction A basic prob
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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 and 16:
X is to use one-parameter subgroups
Page 17 and 18: Chapter 2 Toric varieties and group
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: on varieties discussed in this sect
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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