Algebraic Homogeneous Spaces and Invariant Theory
The invariant theory of non-reductive groups has its roots in the 19th century but has seen some very interesting developments in the past twenty years. This book is an exposition of several related topics including observable subgroups, induced modules,
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1673
Lecture Nates in Mathematics Editors: A. Dold, Heidelberg F. Takens. Groningen
1673
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Frank D. Grosshans
Algebraic Homogeneous Spaces and Invariant Theory
Springer
Author Frank D. Grosshans Department of Mathematics Wester Chester University of Pennsylvania West Chester, PA 19383, USA E-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Grosshans, Frank D.:
Algebraic homogeneous spaces and invariant theory / Frank D. Grosshans. - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo: Springer, 1997 (Lecture notes in mathematics; 1673) ISBN 3-540-63628-5
Mathematics Subject Classification (1991): l3A50, 14L30, l5A72, 20G 15 ISSN 0075-8434 ISBN 3-540-63628-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10553372 46/3142-543210 - Printed on acid-free paper
Contents
Introduction . . . . . . . . . . . . . . ..
1
Chapter One - Observable Subgroups
§ 1. §2. §3. §4.
Stabilizer Subgroups ... . . . . . . . . . Equivalent Conditions .. . . Observable Subgroups of Reductive Groups . Finite Generation of k[G/H]. . . . . . . Appendix: On Valuation Rings §5. Maximal Unipotent Subgroups Bibliographical Note .
. 5 10 14 19 23 27 32
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 . . . . . . . . . . §1O. The Theorems of Roberts and Weitzenbock §11. Geometric Examples . A. Multiplicity-free actions . B. Affine Geometry . C. Invariants of the Orthogonal Group D. Euclidean Geometry E. Hilbert's Example .
33 36 41 43 46
49 53 59 59
62 63 65
67
Chapter Three - Invariants of Maximal Unipotent Subgroups . §12. The Representations E(w) §13. An Example: The General Linear Group A. Straightening . . B.
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