Equivariant Embeddings of Algebraic Groups

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equivalent to one-parameter subgroups γ 1 ′ , γ′ 2 ∈ X ∗(T ) and δ i = [γ i ′]. By part 4, γ′ 1 , γ′ 2 ∈ Γ(X, x 0) as well. Then γ 1 ′ , γ′ 2 ∈ Γ(X, x 0) ∩ X ∗ (T ), which is the strongly convex rational polyhedral cone associated to the toric variety T ⊂ X by part 3. As strongly convex rational polyhedral cones are convex, the line in X ∗ (T ) joining γ 1 and γ 2 is contained in Γ(X, x 0 ) ∩ X ∗ (T ), and hence the line in the spherical building joining δ 1 and δ 2 is contained in the image of Γ(X, x 0 ), so this image is semi-convex. It is convex by part 2, which implies no pair of antipodal points of the building can belong to the image of Γ(X, x 0 ). Consider the following examples. Example 5 (G viewed as an affine G-embedding). Consider the trivial G-embedding, G ⊂ G with base point e. Here Γ(G, e) = {γ ∈ X ∗ (G) : lim t→0 γ(t) exists in G} = {ε}, where ε denotes the trivial one-parameter subgroup, t ↦→ e, of G. Example 6 (One variety as two distinct embeddings). Let B = {( a −1 b 0 a ) : a, b ∈ k, a ≠ 0 } , which is not a reductive group. Then B → X = A 2 via ( a −1 b 0 a ) ↦→ (b, a) is a B-embedding. Here ⎧ ⎛ ⎞ ⎫ ⎪⎨ ⎪⎬ Γ(X, (0, 1)) = ⎪⎩ γ(t) = ⎜ ⎝ t−n 0 ⎟ ⎠ : n ∈ N 0 . 0 t n ⎪ ⎭ However, B → Y = A 2 given by ( a −1 b 0 a ) ↦→ (a −1 , a −1 b) is another B-embedding into A 2 , but ⎧ ⎛ ⎞ ⎫ ⎪⎨ ⎪⎬ Γ(Y, (1, 0)) = ⎪⎩ γ(t) = ⎜ ⎝ t−n c(t n − t −n ) ⎟ ⎠ : c ∈ k, −n ∈ N 0 . 0 t n ⎪ ⎭ Therefore, isomorphism of embeddings is a finer relation than isomorphism of varieties. As a result, our notation Γ(X, x 0 ) may seem a bit clumsy as it does not indicate the manner in which B is embedded in X, but this will be clear from the context whenever it is used. Example 7 (Γ(X, x 0 ) for the group embedding associated to a toric variety). Suppose T is a maximal torus of G and T σ is an affine toric variety for T . Then G × T T σ is an affine embedding of G. Clearly σ ⊆ Γ(G × T T σ , [e, e]), as σ = Γ(G × T T σ , [e, e]) ∩ X ∗ (T ). For each parabolic subgroup P of G containing T , let ∆ P (T ) = {γ ∈ X ∗ (T ) : P (γ) ⊇ P }. Then Γ(G × T T σ , [e, e]) = 42
⋃ P ⊃T P • (σ ∩ ∆ P (T )) is a simplicial decomposition of Γ(G × T T σ , [e, e]) ⊂ X ∗ (G). We remark that this is a finite simplicial decomposition, for there are only finitely many parabolic subgroups of G containing T . (The Borel subgroups of G that contain T are all conjugate to a fixed one by elements of the Weyl group, W (T, G), which is finite, and the set of parabolic subgroups of G containing a given Borel subgroup B ⊃ T are indexed by subsets of the set of simple roots ∆(T, B) relative to B.) For instance, consider the GL 2 -embedding GL 2 × D 2 A 2 . Here σ = {γ m,n (t) = ( t m 0 0 t n ) : m, n ∈ N 0 } ⊂ X ∗ (D 2 ). There are only three parabolic subgroups of GL 2 containing D 2 , namely B + = { ( ) a b 0 d }, B − = { ( ) a 0 c d } and GL2 itself. Then ∆ B +(D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m ≥ n}, ∆ B −(D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m ≤ n}, ∆ GL2 (D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m = n}, so σ ∩ ∆ B +(D 2 ) = {γ m,n : m ≥ n ≥ 0}, σ ∩ ∆ B −(D 2 ) = {γ m,n : n ≥ m ≥ 0} and σ ∩ ∆ GL2 (D 2 ) = {γ m,n : m = n ≥ 0}. Now B + • (σ ∩ ∆ B +(D 2 )) = {t ↦→ ( B − • (σ ∩ ∆ B −(D 2 )) = {t ↦→ ( ) t m bt n (1−t m−n ) 0 t n ) t m 0 ct m (1−t n−m ) t n GL 2 • (σ ∩ ∆ GL2 (D 2 )) = {t ↦→ ( t m 0 0 t n ) : m = n ≥ 0}, : m ≥ n ≥ 0}, : n ≥ m ≥ 0}, with limits ⎧ [( 1 b 0 1 [( 1 b 0 1 ) , (0, 0) ] ) , (0, 1) ] m > n > 0 m > n = 0 lim [(p • γ m,n(t)) · [e, (1, 1)]] = t→0 ⎪⎨ ⎪⎩ [( 1 c 0 1 ) , (0, 0)] n > m > 0 [( 1 c 0 1 ) , (1, 0)] n > m = 0 [( 1 0 0 1 ) , (0, 0)] m = n > 0 [( 1 0 0 1 ) , (1, 1)] m = n = 0 43
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 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: ational polyhedral cone in X ∗ (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

Equivariant Embeddings of Algebraic Groups

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