Algebraic Homogeneous Spaces and Invariant Theory

Introduction . . . . . . . . . . . . . . . . . . . . . . Chapter One - Observable Subgroups 1. Stabilizer Subgroups . . . . . . . . . . . . . . . 2. Equivalent Conditions. . . . . . . . . . . . . . . 3. Observable Subgroups of Reductive Groups . . . . . 4. Finite Generation of kÄG/HÜ. . . . . . . . . . . . Appendix: On Valuation Rings. . . . . . . . . 5. Maximal Unipotent Subgroups. . . . . . . . . . . . Bibliographical Note. . . . . . . . . . . . . . . . . . Chapter Two - The Transfer Principle 6. Induced Modules. . . . . . . . . . . . . . . . . . Appendix: Affine Quotients and induced modules 7. Induced Modules and Observable Subgroups . . . . . Appendix: On a Theorem of F. A. Bogomolov . . 8. Counter-examples . . . . . . . . . . . . . . . . . 9. The Transfer Principle . . . . . . . . . . . . . . 10. The Theorems of Roberts and Weitzenb'ck. . . . . . 11. Geometric Examples . . . . . . . . . . . . . . . . A. Multiplicity-free actions . . . . . . . . B. Affine Geometry . . . . .. . . . . . . . C. Invariants of the Orthogonal Group. . . . D. Euclidean Geometry. . . . . . . . . . . . E. Hilbert's Example. . . . . . . . . . . . Chapter Three - Invariants of Maximal Unipotent Subgroups 12. The Representations E( ) . . . . . . . . . . . . . 13. An Example: The General Linear Group . . . . . . . A. Straightening . . . . . . . . . . . . . . B. U - invariants. . . . . . . . . . . . . . C. Results of K. Pommerening . . . . . . . . 14. The Relationship between A and G AU. . . . . . . . 15. The Algebra grA. . . . . . . . . . . . . . . . . . 16. Finite Generation and U-invariants . . . . . . . . A. Algebras. . . . . . . . . . . . . . . . . B. Modules . . . . . . . . . . . . . . . . . 17. S-varieties. . . . . . . . . . . . . . . . . . . . 18. Flat Deformations and Normality. . . . . . . . . . Bibliographical Note. . . . . . . . . . . . . . . . . . Chapter Four - Complexity 19. Basic Principles . . . . . . . . . . . . . . . . . Appendix: On Quotient Spaces . . .. . .. . 20.