Mathematical Surveys and Monographs Volume 154

Parabolic I Background and General

Andreas Cˇ ap Jan Slovák

American Mathematical Society http://dx.doi.org/10.1090/surv/154 Parabolic Geometries I Background and General Theory

Mathematical Surveys and Monographs Volume 154

Parabolic Geometries I Background and General Theory

Andreas Cˇ ap Jan Slovák

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen, Chair J. T. Stafford Benjamin Sudakov

2000 Subject Classification. Primary 53C15, 53A40, 53B15, 53C05, 58A32; Secondary 53A55, 53C10, 53C30, 53D10, 58J70.

The first author was supported during his work on this book at different times by projects P15747–N05 and P19500–N13 of the Fonds zur F¨orderung der wissenschaftlichen Forschung (FWF). The second author was supported during his work on this book by the grant MSM0021622409, “Mathematical structures and their physical applications” of the Czech Republic Research Intents scheme. Essential steps in the preparation of this book were made during the authors’ visit to the “Mathematisches Forschungsinstitut Oberwolfach” (MFO) in the framework of the Research in Pairs program.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-154

Library of Congress Cataloging-in-Publication Data Cap,ˇ Andreas, 1965– Parabolic geometries / Andreas Cap,ˇ Jan Slov´ak, v. cm. — (Mathematical surveys and monographs ; v. 154) Includes bibliographical references and index. Contents: 1. Background and general theory ISBN 978-0-8218-2681-2 (alk. paper) 1. Partial differential operators. 2. Conformal . 3. Geometry, Projective. I. Slov´ak, Jan, 1960– II. Title. QA329.42.C37 2009 515.7242–dc22 2009009335

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Preface vii

Part 1. Background 1 Chapter 1. Cartan geometries 3 1.1. Prologue — a few examples of homogeneous spaces 4 1.2. Some background from differential geometry 15 1.3. A survey on connections 35 1.4. Geometry of homogeneous spaces 49 1.5. Cartan connections 70 1.6. Conformal Riemannian structures 112 Chapter 2. Semisimple Lie and Lie groups 141 2.1. Basic structure theory of Lie algebras 141 2.2. Complex semisimple Lie algebras and their representations 160 2.3. Real semisimple Lie algebras and their representations 199

Part 2. General theory 231 Chapter 3. Parabolic geometries 233 3.1. Underlying structures and normalization 234 3.2. Structure theory and classification 290 3.3. Kostant’s version of the Bott–Borel–Weil theorem 339 Historical remarks and references for Chapter 3 360 Chapter 4. A panorama of examples 363 4.1. Structures corresponding to |1|–gradings 363 4.2. Parabolic contact structures 402 4.3. Examples of general parabolic geometries 426 4.4. Correspondence spaces and twistor spaces 455 4.5. Analogs of the Fefferman construction 478 Chapter 5. Distinguished connections and curves 497 5.1. Weyl structures and scales 498 5.2. Characterization of Weyl structures 517 5.3. Canonical curves 558

v vi CONTENTS

Appendix A. Other prolongation procedures 599 Appendix B. Tables 607 Bibliography 617 Index 623 Preface

The roots of the project to write this book originated in the early nineteen– nineties, when several streams of mathematical ideas met after being developed more or less separately for several decades. This resulted in an amazing interac- tion between active groups of working in several directions. One of these directions was the study of conformally invariant differential operators re- lated, on the one hand, to Penrose’s twistor program (M.G. Eastwood, T.N. Bailey, R.J. Baston, C.R. Graham, A.R. Gover, and others) and, on the other hand, to hypercomplex analysis (V. Souˇcek, F. Sommen, and others). It turned out that, via the canonical Cartan connection or equivalent data, conformal geometry is just one instance of a much more general picture. Over the years we noticed that the foundations for this general picture had already been developed in the pioneering work of N. Tanaka. His work was set in the language of the equivalence problem and of differential systems, but independent of the much better disseminated de- velopments of the theory of differential systems linked to names like S.S. Chern, R. Bryant, and M. Kuranishi. While Tanaka’s work did not become widely known, it was further developed, in particular, by K. Yamaguchi and T. Morimoto, who put it in the setting of filtered manifolds and applied it to the geometric study of systems of PDE’s. A lot of input and stimulus also came from Ch. Fefferman’s work in and geometric function theory, in particular, his parabolic invariant theory program and the relation between CR–structures and conformal structures. Finally, via the homogeneous models, all of these studies have close relations to various parts of representation theory of semisimple Lie groups and Lie algebras, developed for example by T. Branson and B. Ørsted. Enjoying the opportunities offered by the newly emerging International Erwin Schr¨odinger Institute for (ESI) in Vienna as well as the long lasting tradition of the international Winter Schools “Geometry and Physics” held every year in Srn´ı, Czech Republic, the authors of this book started a long and fruitful collaboration with most of the above mentioned people. Step by step, all of the general concepts and problems were traced back to old masters like Schouten, Veblen, Thomas, and Cartan, and a broad research program led to a conceptual understanding of the common background of the various approaches developed more recently. In the late nineties, the general version of the invariant for a vast class of geometrical structures extended the tools for geometric analy- sis on homogeneous vector bundles and the direct applications of representation theory expanded in this way to situations involving curvatures. In particular, the celebrated Bernstein–Gelfand–Gelfand resolutions were recovered in the realm of general parabolic geometries by V. Souˇcek and the authors and the cohomological substance of all these constructions was clarified.

vii viii PREFACE

The book was written with two goals in mind. On one hand, we want to provide a (relatively) gentle introduction into this fascinating world which blends and geometry, Lie theory and geometric analysis, geometric intuition and categorical thinking. At the same time, preparing the first treatment of the general theory and collection of the main results on the subject, gave us the ambition to provide the standard reference for the experts in the area. These two goals are reflected in the overall structure of the book which finally developed into two volumes. The second volume will be called “Parabolic Geometries II: Invariant Differential Operators and Applications”.

Contents of the book. The basic theme of this book is canonical Cartan connections associated to certain types of geometric structures, which immediately causes peculiarities. Cartan connections and, more generally, various types of absolute parallelisms certainly played a central role in Cartan’s work on differential geometry and they still belong to the basic tools in several geometric approaches to differential equa- tions. However, in the efforts in the second half of the last century to put Cartan’s ideas into the conceptual framework of fiber bundles, the main part was taken over by principal connections. The isolated treatments in the books by Kobayashi and Sharpe stopped at alternative presentations of the quite well–known conformal Rie- mannian and projective structures. As a consequence, even basic facts on Cartan connections are neither well represented in the standard literature on differential geometry nor well known among many people working in the field. For this book, we have collected some general facts about Cartan connections on principal fiber bundles in Section 1.5. While this is located in the “Background” part of the book because of its nature, quite a bit of this material will probably be new to most readers. The other peculiar consequence of the approach is that the geometric structures we study display strong similarities in the picture of Cartan connections, while they are extremely diverse in their original descriptions. Therefore, in large parts of the book we will take the point of view that the Cartan picture is the “true” description of the structures in question, while the original description is obtained as an underlying structure. This point of view is justified by the results on equivalences of categories between Cartan geometries and underlying structures in Section 3.1, which are among the main goals of this volume. This point of view will also be taken in the second volume, in which the Cartan connection (or some equivalent data) will be simply considered as an input. Let us describe the contents of the first volume in more detail. The techni- cal core of the book is Chapter 3. In Section 3.1 we develop the basic theory of parabolic geometries as Cartan geometries and prove the equivalence to underlying structures in the categorical sense. This is done in the setting of |k|–gradings of semisimple Lie algebras, thus avoiding the use of structure theory and representa- tion theory. The structure theory is brought into play in Section 3.2 to get more detailed information on the applicability of the methods developed before. Section 3.3 contains an exposition and a complete proof of Kostant’s version of the Bott– Borel–Weil Theorem, which is needed to verify the cohomological conditions that occur in several places in the theory, and proves to be extremely useful later, too. In Chapter 4, the general results of Chapter 3 are turned into explicit de- scriptions of a wide variety of examples of geometries covered by our methods. PREFACE ix

In particular, we thoroughly discuss the geometries corresponding to |1|–gradings (which can be described as classical first order G–structures) and the parabolic contact geometries, which have an underlying contact structure. In Sections 4.4 and 4.5, we discuss two general constructions relating geometries of different types, the construction of correspondence spaces and twistor spaces, and analogs of the Fefferman construction. The developments in Chapter 5 admit two interpretations. On the one hand, via the notion of Weyl structures, we associate to any parabolic geometry a class of distinguished connections and we define classes of distinguished curves. On the other hand, the data associated to a Weyl structure offer an equivalent description of the canonical Cartan connection in terms of objects associated to the underlying structure. In this way, one also obtains a more explicit description of the canon- ical Cartan connections. Throughout Sections 5.2 and 5.3 we also discuss various applications of the theory developed in the book. The first part of the book (Chapters 1 and 2) provides necessary background and motivation. Chapter 1 is general and rather elementary and should be digestible and enjoyable even for newcomers. Here Cartan’s concept of “curved analogs” of Klein’s homogeneous spaces and also the related general calculi are explained us- ing the effective general language of Lie groups and Lie algebras but no structure theory. As mentioned before, some of the material presented in this discussion is not easy to find in the literature. Section 1.6 contains an explicit and elementary treatment of conformal (pseudo)–Riemannian structures. Apart from motivating further developments, this also indicates clearly that a deep understanding of the algebraic structure of the algebras and groups of in question is the key to further progress. This naturally leads to Chapter 2, which contains background material on semisimple Lie algebras and Lie groups. While the material we cover in this chapter is certainly available in book form in many places, there are some unusual aspects. The main point is that, apart from the complex theory, we also discuss the structure theory and representation theory of real semisimple Lie alge- bras. The real theory is typically scattered in the textbooks among the advanced topics and hence rather difficult to learn quickly elsewhere. In this way, the first part of the book makes the whole project more or less self–contained. In addi- tion, it should be of separate interest as well. As an important counterpart to the theory developed in Chapter 2 we provide tables containing the central structural information on semisimple Lie algebras in Appendix B. The second volume will be devoted to invariant differential operators for par- abolic geometries, in particular, the technique of BGG–sequences, and several ap- plications. While the links of the Cartan geometry to the more easily visible and understandable underlying structures are among the main targets of the first vol- ume, the second one will treat the Cartan connections as given abstract data. This will further underline the algebraic and cohomological character of the available tools and methods.

Suggestions for reading. We have tried to design the book in a way which allows fruitful reading for people with different interests. Readers interested in one or a few specific examples of the geometries covered by the general theory could start reading the parts of the fourth and fifth chapters devoted to the structures in question, and return to the earlier chapters to get background or general results and xPREFACE concepts as needed. Apart from the well–known conformal and projective struc- tures, the book contains extensive material on almost Grassmannian structures, al- most quaternionic geometries, CR–geometries and Lagrangean contact geometries, quaternionic contact geometries, low–dimensional distributions, path geometries, and many others. This includes part of their twistor theory, correspondence spaces and further functorial constructions. Readers familiar with differential geometry and Lie theory, who are interested in the general approach to parabolic geometries might prefer to look at the “Pro- logue” 1.1 in order to get a sense for the typical examples of the structures in question, inspect briefly the generalities on Cartan connections in 1.5 and then be- gin a serious reading straight from Chapter 3. If necessary, it should be possible to find background material, concepts and technicalities quickly in Chapters 1 and 2. Finally, both Chapters 1 and 2 are also intended to be useful as a broader introduction to the subject and have been used successfully as underlying material for various graduate courses, both in Vienna and Brno. Thus we also believe that readers at the graduate student level may enjoy reading the book in the order it is written, with possible glimpses backward and forward for illustrations of the general considerations as they go. Acknowledgement. Although we heard that delays are common with math- ematical books, the work on this one certainly has taken much longer than anyone expected, and we are grateful to the publisher for his patience and support during all of these years. We would also like to mention here many of our collaborators and colleagues who contributed countless suggestions and comments during the ten years of the development of this project. Our particular thanks are due to those colleagues and students of ours who attended our seminars in Vienna and Brno, where most of the contents of this first volume of our book was read, presented and discussed in detail — the project could hardly have been accomplished without their efforts.

Vienna and Brno, August 20, 2008

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P –frame bundle, 274, 384, 421, 605 Kostant’s version, 351 normal, 277 Bruhat decomposition, 333 W p , 325 Bruhat order, 324 Wp ,325 bundle of scales, 503 Σ–height, 292 conformal, 124 α–string through β,167 r–determined, 562 canonical curves |1|–grading, 296 in Cartan geometries, 110 |k|–grading, 238 on homogeneous spaces, 69 preferred parametrization, 568 absolute derivative, 38 Carnot group, 267 action, 24 Cartan connection, 71 effective, 25 conformal, 128 free, 25 Cartan decomposition, 202 transitive, 25 global, 206 adjoint action, 20 Cartan geometry, 71 admissible Cartan involution, 202 family of curves, 110 global, 206 affine extension, 47 Cartan matrix, 169 AHS structure, 137 Cartan product, 186 algebraic bracket Cartan subalgebra of adjoint tractors, 85 θ–stable, 213 ambient connection complex, 162 for contact projective structures, 582 maximally compact, 213 for projective structures, 528 maximally noncompact, 213 ambient metric, 531 real, 200, 213 associated bundle, 28 Cartan’s criteria, 150 associated graded Casimir element, 190 vector space, 235 Casimir operator atlas, 15 of a representation, 152 automorphism, 163 center infinitesimal, 97 of a Lie algebra, 144 inner, 163 of the universal enveloping algebra, 192 central character, 192 Bianchi identity centralizer, 163 general, 88 chains,574,584,591 reductive, 90 character, 194 biholomorphism, 32 chart, 15 Borel fixed point theorem, 307 Christoffel symbols, 39 Borel subalgebra, 291 cocycle standard, 184, 291 of transition functions, 27 Borel–Moore homology, 337 ideal, 142 Borel–Weil theorem, 185 complete reducibility, 152 Bott–Borel–Weil theorem, 357 complexification

623 624 INDEX

of a Lie algebra, 147 densities of a representation, 148, 227 conformal, 135, 371 cone, 471 derivation cone structure, 471 inner, 153 conformal circles, 560 of a Lie algebra, 153 conformal holonomy, 527 with values in a representation, 158 conformal structure derived series, 142 almost Einstein, 526 development, 108 anti–self–dual, 383 diffeomorphism, 15 generalized, 471 differential form, 18 self–dual, 383 direct sum connection of Lie algebras, 145 affine, 42 distribution, 18 distinguished, 365 bracket generating, 429 general, 37 horizontal, 36 induced, 40 integrable, 18, 19 invariant linear, 62 smooth, 18 invariant principal, 57 dominant, 176 , 170 linear, 35 extended, 178 on a G–structure, 46 partial affine, 48 Einstein metric, 525 partial linear, 47 Engel’s theorem, 143 principal, 38 equivariant, 145 projective equivalence of, 384 Euclidean group, 47 special symplectic, 555 exponential map, 20 contact connection, 405 extension functor, 106 contact form, 403 exterior absolute differential, 38 contact grading, 298 exterior derivative, 18 contact structure, 403 contact torsion, 422 Fefferman construction coordinates classical, 479 normal, 45 general, 108, 478 correspondence, 467 Fefferman space, 478 correspondence space, 99 fiber bundle, 25 cotangent bundle, 17 homogeneous, 50 cotangent space, 17 fibered manifold, 25 Cotton–York tensor fibered morphism, 25 |1|–graded case, 365 filtered vector space, 235 conformal, 131 flow, 17 of a Weyl form, 537 foliation, 19 covariant derivative, 35 frame bundle, 27 adapted, 285 covariant exterior derivative, 37 noncommutative, 604 CR–structure, 414, 552 orthonormal, 113 codimension two frame form, 274 elliptic, 445 Freudenthal multiplicity formula, 194 hyperbolic, 445 Frobenius reciprocity higher codimension, 443 algebraic, 160 curvature geometric, 54 harmonic, 265 Frobenius theorem, 19 of a general connection, 37 fundamental derivative of a linear connection, 36 definition, 86 of a principal connection, 39 fundamental vector field, 25 of a Weyl form, 518 curvature form, 71 G–structure, 45, 113 curvature function, 71 generalized flag variety curve complex, 302 quaternionic, 573 real, 314 INDEX 625 generalized geodesics, 560 of a Klein geometry, 49 generic distribution Killing form, 149 n n(n+1) rank in dim. 2 , 430 Klein geometry, 49 rank 2 in dim. 5, 431, 493 effective, 49 rank 3 in dim. 6, 430, 494 infinitesimally effective, 49 rank 4 in dim. 7, 434 reductive, 50 grading element, 118, 239 split, 50 grading section, 503 Klimyk’s formula, 197 growth vector, 429 Kostant codifferential, 261, 341 Kostant Laplacian, 263, 343 Harish–Chandra map, 193 Kostant multiplicity formula, 196 Hasse diagram, 325 Kostant partition function, 195 height Kostant’s version of the BBW–Theorem, of a root, 174 351 Heisenberg algebra, 403 quaternionic, 433 Laplacian split quaternionic, 435 conformal, 137 highest weight leaf, 19 theorem of the, 184 Levi decomposition, 156 highest weight vector, 182 Levi factor, 156 holomorphic, 33 Levi subgroup, 242 holonomy, 526 Levi bracket, 251 affine, 534 Lie algebra, 19 |k| exotic, 535, 555 –graded, 238 symplectic, 555 abelian, 141 homogeneity compact, 200 of linear maps, 236 filtered, 237 homogeneous model, 71 nilpotent, 142 homogeneous space, 25 reductive, 144 homomorphism semisimple, 144 of Lie algebras, 141 simple, 144 of Lie groups, 20 solvable, 142 horizontal Lie algebra cohomology differential form, 29 definition, 157 horizontal lift, 36 Lie algebra homology horizontal projection, 37 definition, 157 Lie bracket ideal, 24 of adjoint tractors, 85 in a Lie algebra, 141 of vector fields, 17 immersion, 16 Lie derivative, 31 induced module, 68, 159, 184 , 19 infinitesimal character, 192 complex, 32 infinitesimal flag structure, 248 Lie subalgebra, 141 regular, 251 Lie subgroup, 24 integrability virtual, 24 for CR–structures, 414 Lie’s theorem, 143 invariant differential operator, 65 Liouville theorem, 133 isotropy subgroup, 25 local diffeomorphism, 15 isotypical component, 189, 348 locally flat, 74 Iwasawa decomposition, 209 logarithmic derivative, 21 global, 210 lower central series, 142 lowest form, 190 jet, 30 semi–holonomic, 95 M¨obius space, 116 jet prolongation M¨obius structure, 439 of a bundle, 30 manifold Jordan decomposition, 161 almost complex, 33 complex, 32 kernel filtered, 251 626 INDEX

Maurer–Cartan equation, 21 of complex , 163 Maurer–Cartan form, 21 real form morphism compact, 200 of Cartan geometries, 73 of a Lie algebra, 147, 200 of homogeneous bundles, 50 split, 200 of infinitesimal flag structures, 248 real rank, 310 of representations, 145 real structure, 226 multiplicity Reeb field, 405, 542 of a weight, 181 regular of an irreducible component, 188 element in g,163 representation natural bundle, 29, 79 adjoint, 20, 146 Newlander–Nirenberg theorem, 33 completely reducible, 147 Nijenhuis tensor, 33 complex, 145 for almost CR–structures, 414 conjugate, 226 nilradical, 156 constructions with, 146 normal contragradient, 146 conformal case, 128 derivative of a, 20 dual, 146 one–parameter subgroup, 20 faithful, 146 operator fundamental, 186 conformally invariant, 135 holomorphic, 32, 197 orbit, 25 indecomposable, 147 parabolic geometry index, 228 complex, 280 induced, 54 definition, 244 irreducible, 146 normal, 265 of a Lie algebra, 145 regular, 252 of a Lie group, 20 parabolic subalgebra quaternionic, 228 complex, 291 semisimple, 147 real, 308 simply reducible, 147 parabolic subgroup, 242 unitary, 147 opposite, 267 Rho tensor, 501, 518 | | parallel transport, 38 1 –graded case, 365 partition of unity, 16 conformal, 131 path geometry, 463, 587 Ricci identity Pl¨ucker embedding, 307 general, 88 Poincar´e , 118 reductive, 90 Poincar´e–Birkhoff–Witt theorem, 159 root, 164 principal bundle, 26 compact, 214 holomorphic, 32 positive, 169 homogeneous, 50 restricted, 207, 308 morphism, 26 simple, 169 projective holonomy, 528 root decomposition, 164 prolongation root lattice, 169 algebraic, 114 root reflection, 168 pseudo–, 116 root space, 164 pullback , 168 of natural bundles, 31 of one–forms, 17 Satake diagram, 216 of vector fields, 17 scale, 504 conformal, 124 quadric, 443 scaling element, 503 Schubert cell, 337 Racah’s formula, 197 Schubert variety, 337 radical, 155 Schur’s lemma, 146 rank section, 25 of a distribution, 18 semisimple INDEX 627

element of g, 162 adjoint, 83, 256 Serre relations, 179 almost Grassmannian, 381 sign conformal standard, 523 of a element, 174 projective standard, 527 simple subsystem, 174 tractor connection, 83 soldering form, 42, 518 abstract, 287 of a Weyl structure, 501 transition function, 25 stabilizer, 25 translation, 19 standard generators, 179 twistor correspondence standard parabolic subalgebra conformal, 470 complex, 291 Grasmannian, 469 real, 308 twistor space, 102, 457 structure for almost quaternionic structures, 473 affine, 44 for conformal structures, 477 almost complex, 33, 280 for quaternionic contact structures, 489 almost Grassmannian, 375, 469 almost Lagrangean, 398 unitary trick, 204 almost quaternionic, 394, 473 universal enveloping algebra, 159 almost spinorial, 400 vector bundle, 26 conformal, 116 associated graded, 237 contact projective, 420, 492, 553, 582 filtered, 237 Grassmannian, 380 holomorphic, 32 hypercomplex, 394 homogeneous, 50 Lagrangean contact, 410, 458, 547, 588 homomorphism, 26 partially integrable almost CR, 412 vector field, 17 projective, 10, 383, 458, 492, 584 complete, 17 ambient description, 528 constant, 71 cone description, 528 left invariant, 19 quaternionic contact, 433, 488 right invariant, 20 split quaternionic contact, 435 Verma module, 184 symplectic, 402 generalized, 320 structure group, 26 vertical projection, 37 reduction of, 27 vertical tangent bundle, 29 submanifold, 16 embedded, 16 Webster scalar curvature, 548, 553 immersed, 16 Webster–Ricci curvature, 546, 553 submersion, 16 Webster–Tanaka connection, 544, 553 subrepresentation, 146 weight, 164 subspace p–algebraically integral, 318 invariant, 146 p–dominant, 318 support, 16 algebraically integral, 180 symbol analytically integral, 197 of a differential operator, 66 dominant, 180 symbol algebra, 251 fundamental, 180 , 224 highest, 183 Hermitian, 304 weight lattice, 181 quaternionic, 304 weight space, 164 Weyl chamber tangent bundle, 16 dominant, 176 tangent map, 16 Weyl character formula, 194, 359 tangent space, 16 Weyl connection, 501, 518 torsion, 44 conformal, 120 of a Cartan connection, 85 Weyl curvature, 383 of a Weyl form, 537 |1|–graded case, 365 torsion free conformal, 131 Cartan geometry, 74 Weyl denominator, 195 trace form, 149 Weyl dimension formula, 195 tractor bundle, 83 Weyl form, 518 628 INDEX

normal, 519 Weyl group, 174 affine action, 193 Weyl structure, 499 associated principal connection on G,511 closed, 125, 508 exact, 125, 508 normal, 515

Yamabe operator, 137 Titles in This Series

154 Andreas Capˇ and Jan Slov´ak, Parabolic geometries I: Background and general theory, 2009 153 Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, 2009 152 J´anos Pach and Micha Sharir, Combinatorial geometry and its algorithmic applications: The Alc´ala lectures, 2009 151 Ernst Binz and Sonja Pods, The geometry of Heisenberg groups: With applications in signal theory, optics, quantization, and field quantization, 2008 150 Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang, Finite dimensional algebras and quantum groups, 2008 149 Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, 2008 148 Patrick Dehornoy with Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, 2008 147 David J. Benson and Stephen D. Smith, Classifying spaces of sporadic groups, 2008 146 Murray Marshall, Positive polynomials and sums of squares, 2008 145 Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin, Simple groups of finite Morley rank, 2008 144 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: Techniques and applications, Part II: Analytic aspects, 2008 143 Alexander Molev, Yangians and classical Lie algebras, 2007 142 Joseph A. Wolf, Harmonic analysis on commutative spaces, 2007 141 Vladimir Mazya and Gunther Schmidt, Approximate approximations, 2007 140 Elisabetta Barletta, Sorin Dragomir, and Krishan L. Duggal, Foliations in Cauchy-Riemann geometry, 2007 139 Michael Tsfasman, Serge Vlˇadut¸, and Dmitry Nogin, Algebraic geometric codes: Basic notions, 2007 138 Kehe Zhu, Operator theory in function spaces, 2007 137 MikhailG.Katz, Systolic geometry and topology, 2007 136 Jean-Michel Coron, Control and nonlinearity, 2007 135 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: Techniques and applications, Part I: Geometric aspects, 2007 134 Dana P. Williams, Crossed products of C∗-algebras, 2007 133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006 132 J. P. May and J. Sigurdsson, Parametrized homotopy theory, 2006 131 Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, 2006 130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, 2006 129 William M. Singer, Steenrod squares in spectral sequences, 2006 128 Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painlev´e transcendents, 2006 127 Nikolai Chernov and Roberto Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy Transform, 2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006 TITLES IN THIS SERIES

123 Barbara Fantechi, Lothar G¨ottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck’s FGA explained, 2005 122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005 121 Anton Zettl, Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian Ma and Shouhong Wang, Geometric theory of incompressible flows with applications to fluid dynamics, 2005 118 Alexandru Buium, differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic : L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups I. Structure of strongly quasithin K-groups, 2004 110 Bennett Chow and Dan Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic of quadratic forms and Clifford groups, 2004 108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, second edition, 2003 104 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanr´e, Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Glasner, via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004 99 Philip S. Hirschhorn, Model categories and their localizations, 2003 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

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. Parabolic geometries encompass a very diverse class of geometric structures, including such important examples as conformal, projective, and almost quaternionic structures, hyper- surface type CR-structures and various types

of generic distributions. The characteristic ˇí Sláma feature of parabolic geometries is an equiva- lent description by a Cartan geometry modeled

on a generalized flag manifold (the quotient Photo: © Jir of a semisimple Lie group by a parabolic subgroup). Background on , with a view towards Cartan connections, and on semisimple Lie algebras and their representations, which play a crucial role in the theory, is collected in two introductory chapters. The main part discusses the equiva- lence between Cartan connections and underlying structures, including a complete proof of Kostant’s version of the Bott–Borel–Weil theorem, which is used as an impor- tant tool. For many examples, the complete description of the geometry and its basic invariants is worked out in detail. The constructions of correspondence spaces and twistor spaces and analogs of the Fefferman construction are presented both in general and in several examples. The last chapter studies Weyl structures, which provide classes of distinguished connections as well as an equivalent description of the Cartan connection in terms of data associated to the underlying geometry. Several applications are discussed throughout the text.

For additional information and updates on this book, visit AMS on the Web WWWAMSORGBOOKPAGESSURV  www.ams.org

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