Equivariant Embeddings of Algebraic Groups

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[37] V.L. Popov and É.B. Vinberg, On a class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat 36 (1972), no. 4, 749–763. [38] V.L. Popov and É.B. Vinberg, Invariant theory in Algebraic Geometry, 4, 137–315, Itogi Nauki i Tekhniki, Akad. Nauk. SSSR, Vsesoyuz Inst. Nauchn. i Tekhn. Inform., 1989. [39] G. Rousseau, Immeubles sphériques et théorie des invariants, C.R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 5, A247–A250. [40] T.A. Springer, Linear Algebraic Groups, 2nd Edition, Progress in Mathematics 9, Birkhäuser, 1998. [41] E. Strickland, A vanishing theorem for group compactifications, Math. Ann. 277:1 (1987), 165–171. [42] H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), no. 1, 1–28. [43] H. Sumihiro, Equivariant completion II, J. Math. Kyoto Univ. 15 (1975), no. 3, 573–605. [44] A. Tchoudjem, Cohomologie des fibrés en droites sur les compactifications des groupes réductifs, preprint math.AG/0303125, 37 pages. [45] D.A. Timashev, Classification of G-varieties of complexity 1, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 2, 127–162; English translation: Izv. Math. 61 (1997), no. 2, 363–397. [46] D.A. Timashev, Equivariant compactifications of reductive groups, Mat. Sb. 194 (2003), no. 4, 119–146. [47] J. Tits, On buildings and their applications, Proceedings of the International Congress of Mathematicians, Vancouver, 1974 [48] J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics 386, Springer-Verlag, 1974. [49] É.B. Vinberg, Complexity of action of reductive groups, Funct. Anal. Appl. 20 (1986), 1–11. [50] O. Zariski and P. Samuel, Commutative Algebra, vol. 1 and 2, Graduate Texts in Mathematics 28, 29, Springer-Verlag, 1975. 70
Vita David Charles Murphy was born in Grand Rapids, Michigan on September 29, 1973. In 1996 he graduated Summa Cum Laude from Western Michigan University in Kalamazoo receiving a B.A. in Mathematics. He received his M.S. in Mathematics in 1998 from the University of Illinois at Urbana-Champaign. In 2004 he earned his Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign. Throughout the course of his graduate studies, he was a teaching assistant and a VIGRE pre-doctoral fellow. During this time, he won the Department of Mathematics Teaching Assistant Instructional Award (2001), Delta Sigma Omicron Distinguished Teaching Award (2001), and the Irving Reiner Memorial Award in Algebra (2003). 71
Page 1 and 2:
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
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complement of U in G/H. Remark 1 ([
Page 13 and 14:
G. The wonderful compactification o
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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: [24] B. Iversen, Cohomology of Shea
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Equivariant Embeddings of Algebraic Groups

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