Table Of ContentMathematical
Surveys
and
Monographs
Volume 154
Parabolic Geometries I
Background and General
Theory
ˇ
Andreas Cap
Jan Slovák
American Mathematical Society
Parabolic Geometries I
Background and General
Theory
Mathematical
Surveys
and
Monographs
Volume 154
Parabolic Geometries I
Background and General
Theory
ˇ
Andreas Cap
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 Mathematics Subject Classification. Primary53C15, 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).
ThesecondauthorwassupportedduringhisworkonthisbookbythegrantMSM0021622409,
“Mathematicalstructuresandtheirphysicalapplications”oftheCzechRepublicResearchIntents
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
Pairsprogram.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-154
Library of Congress Cataloging-in-Publication Data
Cˇap,Andreas,1965–
Parabolicgeometries/AndreasCˇap,JanSlov´ak,
v.cm. —(Mathematicalsurveysandmonographs;v. 154)
Includesbibliographicalreferencesandindex.
Contents: 1.Backgroundandgeneraltheory
ISBN978-0-8218-2681-2(alk. paper)
1.Partialdifferentialoperators. 2.Conformalgeometry. 3.Geometry,Projective. I.Slov´ak,
Jan,1960– II.Title.
QA329.42.C37 2009
515(cid:1).7242–dc22 2009009335
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
(cid:1)c 2009bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 141312111009
Contents
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 algebras 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 mathematicians working in several directions. One
of these directions was the study of conformally invariant differential operators re-
lated,ontheonehand,toPenrose’stwistorprogram(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,
viathe canonicalCartanconnection or equivalent data, conformal geometryis 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
workincomplexanalysisandgeometricfunctiontheory,inparticular,hisparabolic
invariant theory program and the relation between CR–structures and conformal
structures. Finally, via the homogeneous models, all of these studies have close
relationstovariouspartsofrepresentationtheoryofsemisimple LiegroupsandLie
algebras, developed for example by T. Branson and B. Ørsted.
Enjoying the opportunities offered by the newly emerging International Erwin
Schro¨dinger Institute for MathematicalPhysics (ESI) in Viennaas wellas 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
fruitfulcollaborationwithmostoftheabovementionedpeople. Stepbystep,allof
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 calculus
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
algebra and geometry, Lie theory and geometric analysis, geometric intuition and
categoricalthinking. Atthesametime,preparingthefirsttreatmentofthegeneral
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 tocertain typesof 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
ideasintothe conceptualframework offiber bundles, themain part wastakenover
by principal connections. The isolated treatments in the books by Kobayashi and
Sharpestoppedatalternativepresentationsofthequitewell–knownconformalRie-
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.
Theotherpeculiarconsequenceoftheapproachisthatthegeometricstructures
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”
descriptionofthestructuresinquestion,whiletheoriginaldescriptionisobtainedas
anunderlyingstructure. Thispointofviewisjustifiedbytheresultsonequivalences
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
parabolicgeometriesasCartangeometriesandprovetheequivalencetounderlying
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.
