http://dx.doi.org/10.1090/surv/098

Moment Maps, Cobordisms, and Hamiltonian Group Actions Mathematical Surveys and Monographs Volume 98

Moment Maps, Cobordisms, and Hamiltonian Group Actions

Victor Guillemin Viktor Ginzburg Yael Karshon

American Mathematical Society EDITORIAL COMMITTEE Peter S. Landweber Tudor Stefan Ratiu Michael P. Loss, Chair J. T. Stafford

2000 Mathematics Subject Classification. Primary 53Dxx, 57Rxx, 55N91, 57S15.

ABSTRACT. The book contains a systematic treatment of numerous symplectic geometry questions in the context of cobordisms of Hamiltonian group actions. The topics analyzed in the book in­ clude abstract moment maps, symplectic reduction, the Duistermaat-Heckman formula, geometric quantization, and the quantization commutes with reduction theorem. Ten appendices cover a broad range of related subjects: proper actions of compact Lie groups, equivariant cohomology, the Atiyah-Bott-Berline-Vergne localization theorem, the Kawasaki Riemann-Roch formula, and a variety of other results. The book can be used by researchers and graduate students working in symplectic geometry and its applications.

Library of Congress Cataloging-in-Publication Data Guillemin, V., 1937- Moment maps, cobordisms, and Hamiltonian group actions / Victor Guillemin, Viktor Ginz- burg, Yael Karshon. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 98) Includes bibliographical references and index. ISBN 0-8218-0502-9 (alk. paper) 1. Symplectic geometry. 2. Cobordism theory. 3. Geometric quantization. I. Karshon, Yael, 1964- II. Ginzburg, Viktor L., 1962- III. Title. IV. Mathematical surveys and monographs ; no. 98. QA665.G85 2002 516.3/6-dc21 2002074590

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 Contents

Chapter 1. Introduction 1. Topological aspects of Hamiltonian group actions 2. Hamiltonian cobordism 3. The linearization theorem and non-compact cobordisms 4. Abstract moment maps and non-degeneracy 5. The quantum linearization theorem and its applications 6. Acknowledgements

Part 1. Cobordism

Chapter 2. Hamiltonian cobordism 1. Hamiltonian group actions 2. Hamiltonian geometry 3. Compact Hamiltonian cobordisms 4. Proper Hamiltonian cobordisms 5. Hamiltonian complex cobordisms

Chapter 3. Abstract moment maps 1. Abstract moment maps: definitions and examples 2. Proper abstract moment maps 3. Cobordism 4. First examples of proper cobordisms 5. Cobordisms of surfaces 6. Cobordisms of linear actions

Chapter 4. The linearization theorem 1. The simplest case of the linearization theorem 2. The Hamiltonian linearization theorem 3. The linearization theorem for abstract moment maps 4. Linear torus actions 5. The right-hand side of the linearization theorems 6. The Duistermaat-Heckman and Guillemin-Lerman-Sternber

Chapter 5. Reduction and applications 1. (Pre-)symplectic reduction 2. Reduction for abstract moment maps 3. The Duistermaat-Heckman theorem 4. Kahler reduction 5. The complex Delzant construction 6. Cobordism of reduced spaces vi CONTENTS

7. Jeffrey-Kirwan localization 82 8. Cutting 84

Part 2. Quantization 87

Chapter 6. Geometric quantization 89 1. Quantization and group actions 89 2. Pre-quantization 90 3. Pre-quantization of reduced spaces 96 4. Kirillov-Kostant pre-quantization 99 5. Polarizations, complex structures, and geometric quantization 102 6. Dolbeault Quantization and the Riemann-Roch formula 110 7. Stable complex quantization and Spinc quantization 113 8. Geometric quantization as a push-forward 117 Chapter 7. The quantum version of the linearization theorem 119 1. The quantization of Cd 119 2. Partition functions 125 3. The character of Q(Cd) 130 4. A quantum version of the linearization theorem 134

Chapter 8. Quantization commutes with reduction 139 1. Quantization and reduction commute 139 2. Quantization of stable complex toric varieties 141 3. Linearization of [Q,R]=0 145 4. Straightening the symplectic and complex structures 149 5. Passing to holomorphic sheaf cohomology 150 6. Computing global sections; the lit set 152 7. The Cech complex 155 8. The higher cohomology 157 9. Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds 159 10. Overview of the literature 162

Part 3. Appendices 165

Appendix A. Signs and normalization conventions 167 1. The representation of G on C°°(M) 167 2. The integral weight lattice 168 3. Connection and curvature for principal torus bundles 169 4. Curvature and Chern classes 171 5. Equivariant curvature; integral equivariant cohomology 172

Appendix B. Proper actions of Lie groups 173 1. Basic definitions 173 2. The slice theorem 178 3. Corollaries of the slice theorem 182 4. The Mostow-Palais embedding theorem 189 5. Rigidity of compact group actions 191 Appendix C. Equivariant cohomology 197 1. The definition and basic properties of equivariant cohomology 197 CONTENTS vii

2. Reduction and cohomology 201 3. Additivity and localization 203 4. Formality 205 5. The relation between HQ and H^ 208 6. Equivariant vector bundles and characteristic classes 211 7. The Atiyah-Bott-Berline-Vergne localization formula 217 8. Applications of the Atiyah-Bott-Berline-Vergne localization formula 222 9. Equivariant homology 226

Appendix D. Stable complex and Spinc-structures 229 1. Stable complex structures 229 2. Spinc-structures 238 3. Spinc-structures and stable complex structures 248

Appendix E. Assignments and abstract moment maps 257 1. Existence of abstract moment maps 257 2. Exact moment maps 263 3. Hamiltonian moment maps 265 4. Abstract moment maps on linear spaces are exact 269 5. Formal cobordism of Hamiltonian spaces 273

Appendix F. Assignment cohomology 279 1. Construction of assignment cohomology 279 2. Assignments with other coefficients 281 3. Assignment cohomology for pairs 283 4. Examples of calculations of assignment cohomology 285 5. Generalizations of assignment cohomology 287

Appendix G. Non-degenerate abstract moment maps 289 1. Definitions and basic examples 289 2. Global properties of non-degenerate abstract moment maps 290 3. Existence of non-degenerate two-forms 294

Appendix H. Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization 301 1. The Hamiltonian cobordism ring and characteristic classes 301 2. Characteristic numbers 304 3. Characteristic numbers as a full system of invariants 305 4. Non-degenerate cobordisms 308 5. Geometric quantization 310

Appendix I. The Kawasaki Riemann-Roch formula 315 1. Todd classes 315 2. The Equivariant Riemann-Roch Theorem 316 3. The Kawasaki Riemann-Roch formula I: finite abelian quotients 320 4. The Kawasaki Riemann-Roch formula II: torus quotients 323

Appendix J. Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman 327 1. The Spin -Dirac operator and the Spinc-quantization 327 2. The summary of the results 329 viii CONTENTS

3. Transversally elliptic operators and their indexes 331 4. Index of the operator Ba 333 5. The model operator 335 6. Proof of Theorem 1 336 Bibliography 339

Index 349 Bibliography

[Ad] J. F. Adams, Lectures on Lie groups, Benjamin Inc., New York, 1969. [Al] C. Albert, Le theoreme de reduction de Marsden-Weinstein en geometrie cosymplectique et de contact, (The Marsden-Weinstein reduction theorem in cosymplectic and contact geometry), J. Geom. Phys. 6 (1989), no. 4, 627-649. [ACG] J. M. Arms, R. H. Cushman, and M. J. Gotay, A universal reduction procedure for Hamiltonian group actions. The geometry of Hamiltonian systems (Berkeley, CA, 1989), 33-51, Math. Sci. Res. Inst. Publ., 22, Springer, New York, 1991. [Ar] V. I. Arnold, Mathematical methods of classical mechanics, Second Edition, Graduate Texts in Mathematics 60, Springer-Verlag, New York-Berlin, 1989. [Atl] M. F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics 400, Springer-Verlag, Berlin, Heidelberg, 1974. [At2] M. F. Atiyah, Elliptic operators, discrete groups and von Neuman algebras, Asterisque 32-33 (1976), 43-72. [At3] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. [AB1] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. Math. 86 (1967), 374-407. [AB2] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. [ABS] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), 3-38. [AM] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison- Wesley, 1969. [ASe] M. F. Atiyah and G. B. Segal, The index of elliptic operators: II, Ann. Math. 87 (1968), 531-545. [ASi] M. F. Atiyah and I. M. Singer, The index of elliptic operators: III, Ann. Math. 87 (1968), 484-530. [Au] M. Audin, The topology of torus actions on symplectic manifolds, Birkhauser, 1991. [AK] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255-354. [Bac] K. Baclawski, Galois connections and the Leray spectral sequence, Adv. Math. 25 (1977), 191-215. [Bak] A. Baker, Some calculations with Milnor hypersurfaces and an application to Ginzburg's symplectic bordism ring, Preprint, 1994, http://www.maths.gla.ac.uk/ ajb/dvi-ps.html. [BL] L. Bates and E. Lerman, Proper group actions and symplectic stratified spaces, Pacific J. Math. 181 (1997), 201-229. [BW] S. Bates and A. Weinstein, Lectures on the geometry of quantization, BerkeLey Math­ ematics Lecture Notes; v. 8, American Mathematical Society; Providence, R.I., 1997. [BGV] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Springer- Verlag, Berlin, 1992. [BVl] N. Berline and M. Vergne, Classes caracteristiques equivariantes. Formule de locali­ sation en cohomologie equivariante, C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), 539-541. [BV2] N. Berline and M. Vergne, Zeros d'un champ de vecteurs et classes caracteristiques equivariantes Duke Math. J. 50 (1983), 539-549. [BV3] N. Berline and M. Vergne, The Chern character of a transversally elliptic symbol and the equivariant index, Invent. Math. 124 (1996), 11-49.

339 340 BIBLIOGRAPHY

[BV4] N. Berline and M. Vergne, L'indice equivariant des operateurs transversalement ellip- tiques, Invent. Math. 124 (1996), 51-101. [BGG] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Structure of representations that are generated by vectors of higher weight, Functional Anal. Appl. 5 (1971), 1-8. [Bi] J.-M. Bismut, Localization formulas, superconnections, and the index theorem for fam­ ilies, Comm. Math. Phys. 103 (1986), 127-176. [BR] R. Blattner and J. Rawnsley, Quantization of the action of XJ(k,l) on ]R2(fc+^), J. Funct. Anal. 50 (1983), 188-214. [Bl] R. J. Blattner, Quantization and representation theory, Proc. Sympos. Pure Math. vol. 26, A. M. S. Providence, RI (1973), pp. 147-165. [Borl] A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homoggenes de groupes de Lie compactes, Ann. Math. 57 (1953), 115-207. [Bor2] A. Borel, Seminar on transformation groups, Annals of Math. Studies, Princeton Uni­ versity Press, 1960. [BH] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces, III, Amer. J. Math. 82 (1960), 491-504. [BU] D. Borthwick and A. Uribe, Almost complex structures and geometric quantization, Math. Res. Lett. 3 (1996), 845-861; Erratum: Math. Res. Lett. 5 (1998), 211-212. [Botl] R. Bott Homogeneous vector bundles, Ann. of Math. (2), 66 (1957), 203-248. [Bot2] R. Bott, The index theorem for homogeneous differential operators, in: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), 1965, pp. 167-186, Princeton Univ. Press, Princeton, N.J. [BT1] R. Bott and L. W. Tu, Differential forms in algebraic topology, Springer-Verlag, 1982. [BT2] R. Bott and L. W. Tu, Equivariant characteristic classes in the Cartan model, Preprint 2001, math.DG/0102001. [BT3] R. Bott and L. W. Tu, Equivariant cohomology and equivariant characteristic numbers of a homogeneous space, Preprint 2001, math.AG/0102013. [BdMG] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies 99, Princeton University Press, Princeton, NJ, 1981. [Brad] P. Bradley, Moment maps and Jacobian modules, Ann. Global Anal. Geom. 18 (2000), no. 2, 129-151. [Bravl] M. Braverman, Vanishing theorems for the half-kernel of a Dirac operator, Preprint 1998, math.DG/9805127. [Bravl] M. Braverman, Vanishing theorems on covering manifolds, arXiv:math.DG/9809144. [Brav3] M. Braverman, New proof of the cobordism invariance of the index, Preprint 2000, math.SP/0011237. To appear in Proc. Amer. Math. Soc. [Brav4] M. Braverman, Index theory of equivariant Dirac operators on non-compact manifolds, Preprint 2000, math-ph/001,1045. [BF] M. Braverman and M. Farber, Novikov type inequalities for differential forms with non­ isolated zeros, Math. Proc. Cambridge Philos. Soc. 122 (1997), 357-375. [Brel] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics 34, Springer-Verlag, Berlin, Heidelberg, 1967. [Bre2] G. E. Bredon, Introduction to Compact Transformation Groups, Pure and Applied Mathematics, 46 Academic Press, New York-London, 1972. [BrV] M. Brion and M. Vergne, On the localization theorem in equivariant cohomology Preprint, 1997, dg-ga/9711003. [BJ] Th. Brocker and K. Janich, Introduction to Differential Topology, Cambridge University Press, 1982. [BBGK] J.-L. Brylinski, R. Brylinski, V. Guillemin, and V. Kac (Eds.) Lie theory and geometry. In honor of Bertram Kostant, Progress in Mathematics 123, Birkhauser Boston, Inc., Boston, MA, 1994. [Can] A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag 2001. [CG] A. Cannas da Silva and V. Guillemin, On the Kostant multiplicity formula for group actions with non-isolated fixed points, Adv. Math. 123 (1996), 1-15. [CKT] A. Cannas da Silva, Y. Karshon, and S. Tolman, Quantization of presymplectic mani­ folds and circle actions, Trans. A. M. S. 352 (2000), 525-552. BIBLIOGRAPHY 341

[CW] A. Cannas da Silva and A. Weinstein, Lectures on geometries models for noncommuta- tive algebras, Berkeley Mathematics Lecture Notes 10, American Math. Soc, 1999. [Carl] H. Cartan, Notions d'algebre dijferentielle; application aux groupes de Lie et aux variete ou opere un group de Lie. In Colloque de Topologie, C.B.R.M, Bruxelles, 1950, 15-27. [Car2] H. Cartan, La transgression dans up groupe de Lie et dans up espace fibre principal. In Colloque de Topologie, C.B.R.M, Bruxelles, 1950, 57-71. [CGS] C. Cieliebak, A. R. Gaio, and D. A. Salamon, J-holomorphic curves, moment maps and invariants of Hamiltonian group actions, Int. Math. Res. Notes 10 (2000), 831-882. [CDM] M. Condevaux, P. Dazord, and P. Molino, Geometrie du moment, Travaux du Seminaire Sud-Rhodanien de Geometrie, I, 131-160, Publ. Dep Math. Nouvelle Ser. B. 88-1, Univ. Claude-Bernard, Lyon, 1988. [CF1] P. E. Conner and E. E. Floyd, Periodic maps which preserve a complex structure, Bull. Amer. Math. Soc. 70 (1964), 574-579. [CF2] P. E. Conner and E. E. Floyd, The relation of cobordism to K-theories, Lect. Notes in Math. Springer-Verlag, Berlin, Heidelberg, New York, 1966. [CF3] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Springer-Verlag, Berlin- Gttingen-Heidelberg, 1964. [Coxl] D. A. Cox, Toric varieties and toric resolutions, in: Resolution of singularities (Ober- gurgl, 1997), 259-284, Prog. Math. 181, Birkhauser, Basel, 2000. [Cox2] D. A. Cox, Recent developments in toric geometry, in: Algebraic geometry — Santa Cruz 1995, 389-436, Proc. Sympl. Pure Math., 62, Part 2, Amer. Math. Soc, Providence, RI, 1997. [Co] T. J. Courant, Dirac manifolds, Trans. A. M. S., 319 (1990), 631-661. [CFKS] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrodinger operators with appli­ cations to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, 1987. [Dan] V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33:2 (1978), 97-154. [Da] I. Daubechies, Coherent states and projective representation of the linear canonical transformations, J. Math. Phys. 21 (1980), 377-1389. [Di] J. Dixmier, Enveloping algebras, Graduate Studies in Mathematics, 11, American Math­ ematical Society, Providence, RI, 1996. [De] T. Delzant, Hamiltoniens periodiques et image convexe de Vapplication moment, Bul­ letin de la Societe Mathematique de France 116 (1988), 315-339. [tD] T. torn Dieck, Bordism of G-manifolds and integrality theorems, Topology 9, 1970, 345-358. [Dc] P. A. M. Dirac, The principles of quantum mechanics, Third Edition, Clarendon Press, Oxford, 1947. [Doll] A. Dold, Partitions of unity in the theory of fibrations, Ann. of Math. 78 (1963), 223- 255. [Dol2] A. Dold, Lectures on Algebraic Topology, Springer-Verlag, New York, 1972. [Dou] G. Douglas, Banach algebra techniques in operator theory Pure and Applied Mathe­ matics, Vol. 49. Academic Press, New York-London, 1972. [Dr] K. S. Druschel, Oriented orbifold cobordism, Pac. J. Math. 164, 1994, 299-319. [Duf] M. Duflo, Construction de representations unitaires d'un groupe de Lie, (Construction of unitary representations of a Lie group), In "Harmonic analysis and group represen­ tations", 129-221, C.I.M.E. 1980, Liguori, Naples, 1982. [DKV] M. Duflo, S. Kumar, and M. Vergne, Sur la cohomologie equivariante des variete differentiates, Asterisque 205, 1993. [DPV] M. Duflo, N. V. Pedersen, and M. Vergne (Eds.), The orbit method in representation theory, Proceedings of the conference held at the University of Copenhagen, Copen­ hagen, August 29-September 2, 1988. Progress in Mathematics, 82. Birkhauser Boston, Inc., Boston, MA, 1990. [Du] J. J. Duistermaat, The heat kernel Lefschetz fixed point formula for the Spin Dirac operator. Birkhauser, Boston, 1996. [DGMW] J. J. Duistermaat, V. Guillemin, E. Meinrenken, and S. Wu, Symplectic reduction and Riemann-Roch for circle actions, Math. Res. Lett. 2 (1995), 259-266. 342 BIBLIOGRAPHY

[DH1] J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology in the sym- plectic form of the reduced phase space, Invent. Math. 69 (1982), 259-268. [DH2] J. J. Duistermaat and G. J. Heckman, Addendum to: On the variation in the coho­ mology in the symplectic form of the reduced phase space. Invent. Math. 72 (1983), 153-158. [DK] J. J. Duistermaat and J. A. C. Kolk, Lie groups. Springer, Berlin, New York, 2000. [DET] C. Duval, J. Elhadad and G. M. Tuynman, The BRS method and geometric quantiza­ tion: some examples, Comm. Math. Phys. 126 (1990), 535-557. [Eh] E. Ehrhart, Sur un probleme de geometrie diophantienne lineaire, I, II, Crelle J. (J. Reine Angew. Math.) 226 (1967), 1-29; 227 (1967), 25-49. [Fe] B. V. Fedosov, Non-abelian reduction in deformation quantization, Lett. Math. Phys., 43 (1998), 137-154. [FF] A. T. Fomenko and D. B. Fuchs, A Course in nomotopic Topology, Nauka, Moscow, 1989 (in Russian). [F] T. Frankel, Fixed points and torsion on Kahler manifolds, Ann. of Math. (2) 70 (1959), 1-8. [Fr] T. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathe­ matics, vol. 25, American Mathematical Society, Providence, Rhode Island, 2000. [Fu] W. Fulton, Introduction to toric varieties, Princeton University Press, 1993. [GaSa] A. R. Gaio and D. A. Salamon, Gromov-Witten invariants of symplectic quotients and adiabatic limits, arXiv:math.SG/0106157. [Ge] H. Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455-464. [Gil] P. B. Gilkey, Invariance, the Heat equation and the Atiyah-Singer index theorem, Pub­ lish or Perish, Washington, 1984. [Ginl] V. L. Ginzburg, Cobordisms of symplectic and contact manifolds, Functional Anal. Appl. 23 (1989), no. 2, 106-110. [Gin2] V. L. Ginzburg, Calculation of contact and symplectic cobordism groups, Topology 31, 1992, 767-773. [Gin3] V. L. Ginzburg, Some remarks on symplectic actions of compact groups, Math. Z. 210 1992, 625-640. [Gin4] V. L. Ginzburg, Momentum mappings and Poisson cohomology, Internat. J. Math. 7 (1996), 329-358. [Gin5] V. L. Ginzburg, Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math. 10 (1999), 977-1010. [GGK1] V. L. Ginzburg, V. Guillemin and Y. Karshon, Cobordism theory and localization for­ mulas for Hamiltonian group actions, Int. Math. Res. Notices 5 (1996), 221-234. [GGK2] V. L. Ginzburg, V. Guillemin, and Y. Karshon, The relation between compact and non-compact equivariant cobordisms, In Tel Aviv Topology Conference: Rothenberg Festschrift, M. Farber, W. Lueck, and S. Weinberger (Editors); Publ. AMS, Contempo­ rary Math. Series, vol. 231 (1999), 99-112. [GGK3] V. L. Ginzburg, V. Guillemin, and Y. Karshon, Assignments and abstract moment maps, J. Diff. Geometry 52 (1999), 259-301. [GM] V. Ginzburg and R. Montgomery, Geometric quantization and no-go theorems, Poisson geometry (Warsaw, 1998), 69-77, Banach Center Publ., 51, Polish Acad. Sci., Warsaw, 2000. [Go] B. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225. [GKM] M. Goretsky, R. Kottwitz, and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131, 1998, 25-83. [GGG] M. J. Gotay, J. Grabowski, and H. B. Grundling, An obstruction to quantizing compact symplectic manifolds, Proc. Amer. Math. Soc. 128 (2000), no. 1, 237-243. [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. [Groe] H. J. Groenewald, On the principles of elementary quantum mechanics, Physics 12 (1946), 405-460. [Grom] M. Gromov, Partial differential relations, Springer-Verlag, 1986. BIBLIOGRAPHY 343

[GHS] M. Gromov, G. Henkin and M. Shubin, Holomorphic L2 functions on coverings of pseudocomplex manifolds, GAFA 8 (1998), 525-585. [GK] M. Grossberg and Y. Karshon, Equivariant index and the moment map for completely integrable torus actions, Adv. Math. 133 (1998), 185-223. [Gru] B. Griinbaum, Convex polytopes, J. Wiley and Sons, 1967. [Guic] A. Guichardet, Cohomologie des groupes topologiques et des algebres de Lie, (Coho- mology of topological groups and Lie algebras), Textes Mathematiques (Mathematical texts), 2, Cedic/Fernard Nathan, Paris, 1980. [Guill] V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian Tn-Spaces, Progress in Mathematics vol. 122, Birkhauser Boston 1994. [Guil2] V. Guillemin, Reduced phase spaces and Riemann-Roch, Lie theory and geometry, 305- 334, Progr. Math., 123, Birkhauser Boston, Boston, MA 1994. [GLS] V. Guillemin, E. Lerman and S. Sternberg, Symplectic Fibrations and Multiplicity Di­ agrams, Cambridge Univ. Press, 1996. [GP] V. Guillemin and A. Pollack, Differential topology, Englewood Cliffs, N.J., Prentice-Hall, 1974. [GS1] V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc. Surveys 14, 1977. [GS2] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, In­ vent. Math. 67, 1982, 491-513. [GS3] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group rep­ resentations, Invent. Math. 67 (1982), 515-538. [GS4] V. Guillemin and S. Sternberg, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982), 344-380. [GS5] V. Guillemin and S. Sternberg, Cohomological properties of the moment mapping, Preprint 1982, unpublished. [GS6] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, 1984. [GS7] V. Guillemin and S. Sternberg, A normal form for the moment map, Differential geomet­ ric methods in mathematical physics, (S. Sternberg, Ed.) Reidel, Dordrecht, Holland, 1984. [GS8] V. Guillemin and S. Sternberg, Super symmetry and equivariant de Rham Theory, Springer, Berlin, 1999. [GU] V. Guillemin and A. Uribe, The Laplace operator on the nth tensor power of a line bundle: eigenvalues which are uniformly bounded in n, Asymptotic Anal. 1 (1988), 105-113. [GR] R. G. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice- Hall, 1965. [GZ1] S. M. Gusein-Zade, The action of a circle on manifolds, (Russian) Mat. Zametki 10 (1971), 511-518. English translation: Math. Notes 10 (1971), 731-734 (1972). [GZ2] S. M. Gusein-Zade, U-actions of a circle and fixed points, Math. USSR Izvestija 5 (1971), no. 5, 1127-1143. [Ham] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 65-222. [HC] Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions. Acta Math. 113 (1965), 241-318, and Discrete series for semisimple Lie groups. II. Explicit determination of the characters. Acta Math. 116 (1966), 1-111. [Hatl] A. Hattori, Spin0-structures and S1-actions, Invent. Math., 48 (1978), 7-31. [Hat2] A. Hattori, S1-actions on unitary manifolds and quasi-ample line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1985), 433-486. [Hat3] A. Hattori, Almost complex toric manifolds and positive line bundles, Homotopy and geometry (Warsaw, 1997), 95-114, Banach Center Publ. 45, Polish Acad. Sci., Warsaw, 1998. [HY] A. Hattori and T. Yoshida, Lifting compact group actions in fiber bundles, Japan. J. Math. 2 (1976), 13-25. [He] G.J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for com­ pact connected Lie groups, Invent. Math. 67 (1982), 333-356. [Hi] N. Higson, A note on cobordism invariance of the index, Topology 30 (1991), 439-443. 344 BIBLIOGRAPHY

[Hs] W. Y. Hsiang, Cohomology theory of topological transformation groups, Springer-Verlag, 1975. [Hu] S.-T. Hu, Homotopy theory. Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959 [Hue] J. Huebschmann, Symplectic and Poisson structures of certain moduli spaces, Pub. Irma-Lille-93-33.X, arXiv:hep-th/9312112vl. [Hus] D. Husemoller, Fibre Bundles, Third Edition, Springer-Verlag, New York, 1994. [JMcCO] S. Jackowski, J. McClure, and R. Oliver, Homotopy classification of self-maps of BG via G-actions. I, II, Ann. of Math. (2), 135 (1992), 183-226, 227-270. [Jef] L. Jeffrey, Extended moduli spaces of flat connections on Riemann surfaces, Math. Annalen 298, 667-692 (1994). [JK1] L. Jeffrey and F. Kirwan, Localization for non-abelian group actions, Topology 34 (1995), 291-327. [JK2] L. Jeffrey and F. Kirwan, On localization and Riemann-Roch numbers for symplectic quotients, Quart. J. Math. Oxford Ser. (2) 47 (1996), 165-185. [JK3] L. Jeffrey and F. Kirwan, Localization and the quantization conjecture, Topology 36 (1997), 647-693. [Jen] C. U. Jensen, Les Foncteurs derives de lim et leurs applications en theorie des modules, Lecture Notes in Mathematics, 274, Springer-Verlag, Berlin, Heidelberg, 1972. [Kal] Y. Karshon, An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc. 116 (1992), 591-605. [Ka2] Y. Karshon, Hamiltonian torus actions, in: Geometry and Physics, Lecture Notes in Pure and Applied Mathematics Series 184, Marcel Dekker, 1996, pp. 221-230. [Ka3] Y. Karshon, Moment maps and non-compact cobordisms, J. Diff. Geometry 49 (1998), 183-201. [Ka4] Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc. 141 (1999), no. 672. [KTl] Y. Karshon and S. Tolman, The moment map and line bundles over pre-symplectic toric manifolds, J. Diff. Geom. 38 (1993), 465-484. [KT2] Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001), 4831-4861. [KT3] Y. Karshon and S. Tolman, Tall complexity one Hamiltonian torus actions, arXiv:math.SG/0202167. [KK] A. Khovanskii and J. M. Kantor, Integral points in convex polyhedra, combinatorial Riemann-Roch and generalized MacLaurin formulae, Inst. Hautes Etudes Sci. Publ. Math. (1992) 932-937. [KKS] D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Cpmm. Pure Appl. Math., 31 (1978), 481-507. [Kawl] T. Kawasaki, The Riemann-Roch theorem for complex V-manifolds. Osaka J. Math. 16 (1979), 151-159. [Kaw2] T. Kawasaki, The index of elliptic operators over V-manifolds. Nagoya Math. J. 84 (1981), 135-157. [Kil] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Russian Math. Surveys, 17 (1962), 53-104. [Ki2] A. A. Kirillov, Elements of the theory of representations Springer-Verlag, Berlin-New York, 1976. [Ki3] A. A. Kirillov, Theorems and problems in functional analysis, Springer-Verlag, New York, 1982. [Ki4] A. A. Kirillov, Geometric quantization, Current problems in mathematics. Fundamental directions, Vol. 4, 1985, 141-178. [Ki5] A. A. Kirillov, Merits and demerits of the orbit method, Bull. Amer. Math. Soc. 36 (1999), 433-488. [Kir] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 1984. [KN] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience Pub­ lishers, New York; vol. I, 1963; vol. II, 1969. [KS] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 1958, 328-466. BIBLIOGRAPHY 345

[Kost] B. Kostant, Quantization and unitary representations, in Modern Analysis and Appli­ cations, Springer Lecture Notes in Mathematics 170, p. 87-107, New York, Springer- Verlag, 1970. [KR] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809. [KU] L. Kopras and A. Uribe, Quantization of symplectic cobordisms, Math. Res. Lett. 6 (1999), 119-129. [Kosz] J. L. Koszul, Sur certains groupes de transformations de Lie, Colloque international du Centre National de la Recherche Scientifique 52 (1953), 137-141. [Kr] I. Kriz, The Z/p-equivariant complex cobordism ring, in: Homotopy invariant algebraic structures (Baltimore, MD, 1998), 217-223, Contemp. Math., 239, Amer. Math. Soc, Providence, RI, 1999. [Lai] S. Lang, Differential manifolds, Addison-Wesley, 1972. [La2] S. Lang, Algebra, second ed., Addison-Wesley, 1984. [LM] H. Lawson and M. Michelson, Spin Geometry, Princeton, 1989. [Lerl] E. Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247-258. [Ler2] E. Lerman, A convexity theorem for torus actions on contact manifolds, Preprint 2000; arXiv:math.SG/0012017. [Ler3] E. Lerman, Contact toric manifolds, arXiv:math.SG/0107201. [LMTW] E. Lerman, E. Meinrenken, S. Tolman and C. Woodward, Non-abelian convexity by symplectic cuts, Topology 37 (1998), 245-259. [LT1] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), 4201-4230. [LT2] E. Lerman and S. Tolman, Intersection cohomology of S1 symplectic quotients and small resolutions, Duke Math. J. 103 (2000), 79-99. [LW] E. Lerman and C. Willett, The topological structure of contact and symplectic quotient, arXiv:math.SG/0008178. [Lew] D. Lewis, Lagrangian block diagonalization, J. Dyn. Diff. Eq. 4 (1992), 1—41. [Lo] F. Loose, Reduction in contact geometry, J. Lie theory 11 (2001), 9-22. [Mai] B. Malgrange, Ideals of differentiate functions, Oxford University Press, 1966. [Manl] L. Mantini, An integral transform in L2-cohomology for the ladder representation of U(p,q), J. Funct. Anal. 60 (1985), 211-242. [Man2] L. Mantini, An L2-cohomology construction for unitary highest weight modules for U(p,q), Trans. Amer. Math. Soc. 323 (1991), 583-603. [Marl] C. M. Marie, Modele d'action hamiltonienne d'un groupe de Lie sur une variete sym- plectique, Rendiconti del Seminario Matematico 43 (1985), 227-251, Universitae Po- litechnico, Torino. [MSLP] J. E. Marsden, J. C. Simo, D. L^wis and T. A. Posbergh, Block diagonalization and the energy-momentum method. Dynamics and control of multibody systems (Brunswick, ME, 1988), 297-313, Contemp. Math., 97, Amer. Math. Soc, Providence, RI, 1989. [MW] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130. [Marti] S. K. Martin, Transversality theory, cobordisms, and invariants of symplectic quotients, Preprint 2000, arXiv:math.SG/0001001, to appear in Annals of Mathematics. [Mart2] S. K. Martin, Symplectic quotients by a nonabelian group and by its maximal torus, Preprint 2000, arXiv:math.SG/0001002, to appear in Annals of Mathematics. [Masl] W. S. Massey, Obstructions to the existence of almost complex structures, Bull. Amer. Math. Soc. 67 (1961), 559-564. [Mas2] W. S. Massey, Homology and cohomology theory. An approach based on Alexander- Spanier cochains, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46. Marcel Dekker, Inc., New York-Basel, 1978. [Mas3] W. S. Massey, Singular homology theory, Graduate Texts in mathematics 70, Springer- Verlag, 1980. [Masu] M. Masuda, Unitary toric manifolds, multi-fans, and equivariant index, Tohoku Math. J. (2) 51 (1999), no. 2., 237-265. [Mat] E. Matsui, An equivariant Liapunov stability test and the energy-momentum-Casimir method, Preprint 2001, math.DG/0107086. 346 BIBLIOGRAPHY

[May] J. P. May, et. al. Equivariant homotopy and cohomology theory, CBMS Regional Con­ ference Series in Mathematics, Vol. 91. AMS Publications, Providence, 1996. [McDl] D. McDuff, Applications of convex integration to symplectic and contact geometry Ann. Inst. Fourier (Grenoble) 37 (1987), 107-133. [McD2] D. McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), 149-160. [McDSa] D. McDuff and D. Salamon, Introduction to symplectic geometry, Clarendon Press, Oxford, 1995. [Menl] E. Meinrenken, On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc. 9 (1996), 373-389. [Men2] E. Meinrenken, Symplectic surgery and the Spinc-Dirac operator, Adv. Math. 134 (1998), 240-277. [MeSj] E. Meinrenken and R. Sjamaar, Singular reduction and quantization Topology, 38 (1999), 699-762. [MW1] E. Meinrenken and C. Woodward, Hamiltonian loop group actions and Verlinde factor­ ization, J. Diff. Geom. 50 (1998), 417-469. [MW2] E. Meinrenken and C. Woodward, Cobordism for Hamiltonian loop group actions and flat connections on the punctured two-sphere, Math. Z. 231 (1999), 133-168. [MW3] E. Meinrenken and C. Woodward, Moduli spaces of flat connections on 2-manifolds, cobordism, and Witten's volume formulas, Advances in Geometry, 271-295, Progr. Math., 172, Birkhauser Boston, Boston, MA, 1999. [Metl] D. S. Metzler, Topological Invariants of Symplectic Quotients, Ph.D. thesis, M.I.T. June 1997. [Met2] D. S. Metzler, A wall-crossing formula for the signature of symplectic quotients, Trans. Amer. Math. Soc. 352 (2000), no. 8, 3495-3521. [Met3] D. S. Metzler, A K-theoretic note on geometric quantization, Manuscripta Mathematica 100 (1999), no. 3, 277-289. [MiSt] J. W. Milnor and J. D. Stashef, Characteristic classes, Princeton University Press, 1974. [Moe] I. Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Mathematics, 1616, Springer-Verlag, Berlin, Heidelberg 1995. [Mor] J. Morava, Cobordisms of symplectic manifolds and asymptotic expansions, Tr. Mat. Inst. Steklova 225 (1999), Solitony Geom. Topol. na Perekrest., 276-283. [Mos] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. [Most] G. D. Mostow, Equivariant embeddings in Euclidean space, Ann. of Math. (2) 65 (1957), 432-446. [Mul] I. Mundet, Yang-Mills-Higgs theory for symplectic fibrations, Ph.D. Thesis, Madrid, April 1999, arXiv:math.SG/9912150. [Ni] L. I. Nicolaescu, On the cobordism invariance of the index of Dirac operators, Proc. Amer. Math. Soc, 125 (1997), 2797-2801. [NN] L. Nirenberg and A. Newlander, Complex analystic coordinates in almost complex man­ ifolds, Ann. Math. 3 (1957), 391-404. [Od] T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik, Springer, 1988. [Ol] R. Oliver, Vector bundles over classifying spaces, Proceedings of the International Con­ gress of Mathematicians, vol. II (Berlin, 1998). Doc. Math. 1998, extra vol. II, 483-492. [Pal] R. S. Palais, Imbedding of compact, differentiate transformation groups in orthogonal representations, J. Math. Mech. 6 (1957), 673-678. [Pa2] R. S. Palais, Equivalence of nearby differentiate actions of a compact group, Bull. Amer. Math. Soc. 67 (1961), 362-364. [Pa3] R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. Math. 73 (1961), 295-323. [PS] R. S. Palais and T. E. Stewart, Deformations of compact differentiable transformation groups Amer. J. Math. 82 (1960) 935-937. [Pari] P.-E. Paradan, The moment map and equivariant cohomology with generalized coeffi­ cients, Topology 39 (2000), 401-444. [Par2] P.-E. Paradan, Localization of the Riemann-Roch character, J. Funct. Anal., 187 (2001), 442-509. BIBLIOGRAPHY 347

[Par3] P.-E. Paradan, Spinc-quantization and the K-multiplicities of the discrete series, Preprint 2001, math.DG/0103222. [Pr] E. Prato, Convexity properties of the moment map for certain non-compact manifolds, Comm. Anal. Geom. 2 (1994), 267-278. [PW] E. Prato and S. Wu, Duistermaat-Heckman measures in a non-compact setting, Com- positio Math. 94 (1994), no. 2, 113-128. [Pu] L. Pukanszky, Unitary representations of solvable Lie groups, Ann. Sci. Ecole Norm. Sup. (4) 4 (1971), 457-608. [RR] P. L. Robinson and J. H. Rawnsley, The metaplectic representation, Mpc structures and geometric quantization, Mem. Amer. Math. Soc. 81 (1989), no. 410. [RS] M. Reed and B. Simon, Methods of modern mathematical physics IV. Analysis of op­ erators, Academic Press, London, 1978. [Ri] I. Mundet I Riera, Lifts of smooth group actions to line bundles, Preprint 2000, math.DG/0002123. [RS] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer- Verlag, New York, 1972. [Rudi] W. Rudin, Functional analysis, McGraw-Hill, Inc., New York, second edition, 1991. [Rudy] Yu. B. Rudyak, On Thorn spectra, orientability and cobordism, Springer-Verlag, 1998. [SGA4] M. Artin, A. Grothendieck and J. L. Verdier, Theorie des Topos et Cohomologie Etale des Schemas, Lecture Notes in Mathematics, 269, Springer-Verlag, Berlin, Heidelberg, 1972. [Schm] W. Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, in Representation theory and harmonic analysis on semisimple Lie groups, P. J. Sally, Jr. and D. A. Vogan, Jr. (Ed.), Math. Surveys Monographs, 31, 223-286, Amer. Math. Soc, Providence, RI, 1989. [Schw] G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14, 1975, 63-68. [Se] G. Segal, Equivariant K-theory, Publ. Math. IHES, 34 (1968), 129-151. [Sr] J.-P. Serre, Lie algebras and Lie groups, Leet. Notes in Math. 1500, Revised Edition, Springer, Berlin, 1992. [Sh] P. Shanahan, The Atiyah-Singer Index Theorem, Lecture Notes in Mathematics, 638, Springer-Verlag, Berlin, Heidelberg, 1978. [Shi] M. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, New York, 1980. [Sh2] M. Shubin, Semiclassical asymptotics on covering manifolds and Morse inequalities, Geom. Funct. Anal., 6 (1996), 370-409. [sh2] M. Shubin, Spectral theory of elliptic operators on non-compact manifolds, Asterisque 207 (1992), 37-108. [Sh3] M. Shubin, Spectral theory of the Schrodinger operators on non-compact manifolds: qualitative results, in Spectral theory and geometry (Edinburgh, 1998), Cambridge Univ. Press, Cambridge, 1999, pp. 226-283. [SLM] J. C. Simo, D. Lewis, and J. Marsden, The stability of relative equilibria. I. The reduced energy-momentum method, Arch. Rat. Mech. Anal. 115 (1991), 15-59. [Sil] D. Sinha, Computations of Complex Equivariant Bordism Rings, Preprint 1999, math.AT/9910024. [Si2] D. Sinha, Transversality Obstructions and Equivariant Bordism for G = Z/2, Preprint 1999, math.AT/9910025. [Sjl] R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representa­ tions, Ann. of Math. (2) 141 (1995), 87-129. [Sj2] R. Sjamaar, Symplectic reduction and Riemann-Roch formulas for multiplicities, Bull. Amer. Math. Soc. (N.S.) 33 (1996), 327-338. [Sj3] R. Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998), 46-91. [SL] R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction. Ann. of Math. (2) 134 (1991), 375-422. [Ste] N. E. Steenrod, The topology of fibre bundles, Princeton Univ. Press, 1951. [Sg] S. Sternberg, Lectures on differential geometry, Prentice-Hall, NJ, 1964. 348 BIBLIOGRAPHY

[Sto] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, NJ, 1968. [Stu] B. Sturmfels, On vector partition functions, J. Combin. Theory Ser. A 72 (1995), 302- 309. [Sw] R. M. Switzer, Algebraic topology - homotopy and homology, Springer-Verlag, 1975. [Te] C. Teleman, The quantization conjecture revisited, Ann. of Math. (2) 152 (2000), 1-43. [To] S. Tolman, Examples of non-Kahler Hamiltonian torus actions, Invent. Math. 131 (1998), 299-310. [TW] S. Tolman and J. Weitzman, On the cohomology rings of Hamiltonian T-spaces, Preprint 1998, math.DG/9812006. [TZ] Y. Tian and W. Zhang, An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math. 132 (1998), 229-259. [Va] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, 102, Springer-Verlag, New York, 1984. [Vel] M. Vergne, Quantification geometrique et multiplicits, C. R. Acad. Sci. Paris Ser. I Math., 319 (1994), 327-332. [Ve2] M. Vergne, Geometric quantization and equivariant cohomology, First European Con­ gress in Mathematics, vol. 1, Progress in Mathematics 119, Birkhauser, Boston (1994), 249-295. [Ve3] M. Vergne, Equivariant index formulas for orbifolds, Duke Math. J. 82 (1996), 637-652. [Ve4] M. Vergne, Multiplicity formula for geometric quantization, I, II, Duke Math. J. 82 (1996), 143-179 and 181-194. [Ve5] M. Vergne, Quantification geometrique et reduction symplectique, Seminaire Bourbake, March 2001 expose 888. [Vo] D. Vogan, Jr., Unitary representations of reductive Lie groups, Annals of Mathematics Studies, 118, Princeton University Press, Princeton, NJ, 1987. [VHl] L. von Hove, Sur le probleme des relations entre les transformations unitaires de la mecanique quantique et les transformations canonique de la mecanique classique, Acad. Roy. Belgique Bull. CI. Sci. (5)' 37 (1951), 610-620. [VH2] L. von Hove, Sur certaines representations unitaires d'un groupe infini de transforma­ tions, Mem. Acad. Roy. Belgique CI. Sci. 26 (1951), 61-102. [Weil] A. Weinstein, Lectures on symplectic manifolds, Reg. Conf. Ser. Math. 29 (1976). [Wei2] A. Weinstein, Symplectic geometry, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 1-13. [Wei3] A. Weinstein, Poisson geometry, Differential Geom. Appl. 9 (1998), 213-238. [Wei] R. O. Wells., Differential analysis on complex manifolds, Second Edition, Graduate Texts in Mathematics 65, Springer, New York, 1980. [Wt] C. Willett, non-zero contact reduction, arXiv:math.SG/0104080. [Wil] E. Witten, Supersymmetry and Morse theory, J. of Diff. Geom. 17 (1982), 661-692. [Wi2] E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 303-368. [Wd] N. Woodhouse, Geometric Quantization, Oxford University Press, Oxford, 1980. [Ww] C. Woodward, Multiplicity-free Hamiltonian actions need not be Kdhler, Invent. Math. 131 (1998), 311-319. Index abstract moment map, 31 convexity theorem, 21 exact, 32 curvature, 169 Hamiltonian, 32 equivariant, 172 non-degenerate, 289 curvature class, 171 polarized, 34 equivariant, 172 proper, 33 action, 173 Delzant effective, 174 construction faithful, 174 complex, 76 free, 174 symplectic, 75 locally free, 174 polytope, 75 proper, 173 space, 23, 70, 76 additivity techniques, 203 determinant line bundle, 113 alcove, 21, 77 Dirac axioms, 89 assignment, 259 distinguishing cohomology class, 247 ry-polarized, 261 distribution, 103 associated with a moment map, 31 complex, 103 minimal stratum, 263 Liouville, 27 moment, 279 real, 103 assignment cohomology, 279 distributional character, 132 relative, 283 Dolbeault complex, 105 Atiyah-Bott Lefschetz fixed point formula, Duistermaat-Heckman 116 distribution, 27 formula, 60, 222 Bargmann space, 120 function, 70 basic form, 64, 184 integral, 24 Borel construction, 197 measure, 17 Cartan model, 198 polynomial, 72 Chern class, 171, 315 theorem, 69 equivariant, 212, 316 Chevalley's theorem, 200 Ehresmann's lemma, 69 cobordism, 34 equivariant complex oriented, 235 characteristic classes, 212 equivariant, 35 characteristic numbers, 218, 304 Hamiltonian mixed, 304 compact, 25 cohomology, 197 complex, 30 complex line bundles, 214 polarized, 28 de Rham Theorem, 198 proper, 27 differential forms, 198 non-degenerate, 308 homology, 226 proper, 35 Mayer-Vietoris exact sequence, 203 compatible Poincare duality, 223 almost complex structure and symplectic vector bundles, 212 structures, 111 complex and symplectic structures, 103 Fock space, 120 convexity, 294 formality, 206, 290

349 350 INDEX generating vector field, 177 [Q,R] =0, 139 group cohomology, 193 quantization, 104 Guillemin-Lerman-Sternberg formula, 59 Dolbeault, 111 Spinc, 113 Hamiltonian G-manifold, 16 quantization commutes with reduction stable complex, 29 regular values, 140 index theorem, 316 singular values, 141 equivariant, 115, 318 reduced space, 63 reduction Kawasaki invariants, 325 Kawasaki Riemann-Roch formula, 324 for abstract moment maps, 65 Kirwan's epimorphism, 292 symplectic, 64 Riemann-Roch C2 -cohomology, 106 formula, 112 lattice number, 83 group,168 Riemann-Roch theorem weight, 168 equivariant, 316 linearization theorem topological, 118 for abstract moment maps, 51 rigidity, 191 Hamiltonian, 47 ring structure quantum, 134 on set of polarized cobordism classes, 37 Liouville measure, 16, 17 shift formula, 114, 253 lit set, 153 slice theorem, 180 local linearization theorem, 181 Spinc Dirac operator, 113 localization Spinc structure, 113, 238 Jeffrey-Kir wan, 82 bundle equivalence, 240 localization theorem destabilization, 252 Atiyah-Bott-Berline-Vergne, 218, 219 homotopy, 241 Borel's, 204 homotopy classification, 246 moment metric, 242 cone, 22 splitting principle, 208, 315 map, 15 stability theorem, 22 polarized, 27 stabilizer, 174 polytope, 21 infinitesimal, 178 Mostow-Palais embedding theorem, 189 stable complex structure, 229 bundle equivalence, 230 orbit type, 187 equivariant, 229 principal, 188 homotopy, 230 stratification, 188 symplectic cutting, 84 infinitesimal, 188 tame G-manifold, 37 c Pin structure, 244 Todd class, 112, 315 Poisson bracket, 19 equivariant, 317 polarization, 103 polarized function to g*, 34 uniqueness lemmas polarizing vector, 57 for abstract moment maps, 37, 51 polytope for Hamiltonian actions on vector bundles, simple, 75 49 pre-quantization, 91 data, 91 weights, 22, 52 equivariant, 94 isotropy, 22 polarized, 53, 57, 126 line bundle, 90 real, 52 equivariant, 93 principal bundle, 178 push-forward, 219 equivariant, 221 Titles in This Series

98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, .2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, 2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, second edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 TITLES IN THIS SERIES

65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper GofFman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.5 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 5, 2002 40.4 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 4, 1999 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, 1998

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