Equivariant Embeddings of Algebraic Groups

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EQUIVARIANT EMBEDDINGS OF ALGEBRAIC GROUPS BY DAVID CHARLES MURPHY B.A., Western Michigan University, 1996 M.S., University of Illinois at Urbana-Champaign, 1998 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2004 Urbana, Illinois
Abstract We classify embeddings of algebraic groups as open orbits in affine varieties, generalizing results from toric geometry to connected reductive groups. In particular, we show that an embedding is determined by the set of one-parameter subgroups that have a limit in the embedding. We then investigate the structure of such sets of one-parameter subgroups and how their properties reflect properties of the associated embedding. iii
Page 1: c○ Copyright by David Charles Mur
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 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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