D-MODULES, PERVERSE SHEAVES, AND REPRESENTATION THEORY

By W. KULAK

CONTENTS INTRODUCTION 1 PART I D MODULES AND PERVERSE SHEAVES 1 PRELIMINARY NOTIONS 15 1 1 DIFFERENTIAL OPERATORS 15 1 2 D MODULES WARMING UP 17 1 3 INVERSE AND DIRECT IMAGES I 21 1 4 SOME CATEGORIES OF D MODULES 25 1 5 INVERSE IMAGES AND DIRECT IMAGES II 31 1 6 KASHIWARA S EQUIVALENCE 48 1 7 A BASE CHANGE THEOREM FOR DIRECT IMAGES 53 2 COHERENT D MODULES 57 2 1 GOOD LTRATIONS 57 2 2 CHARACTERISTIC VARIETIES SINGULAR SUPPORTS 59 2 3 DIMENSIONS OF CHARACTERISTIC VARIETIES 62 2 4 INVERSE IMAGES IN THE NON CHARACTERISTIC CASE 64 2 5 PROPER DIRECT IMAGES 69 2 6 DUALITY FUNCTORS 70 2 7 RELATIONS AMONG FUNCTORS 76 2 7 1 DUALITY FUNCTORS AND INVERSE IMAGES 76 2 7 2 DUALITY FUNCTORS AND DIRECT IMAGES 77 3 HOLONOMIC D MODULES 81 3 1 BASIC RESULTS 81 3 2 FUNCTORS FOR HOLONOMIC D MODULES 83 3 2 1 STABILITY OF HOLONOMICITY 83 3 2 2 HOLONOMICITY OF MODULES OVER WEYL ALGEBRAS 86 3 2 3 ADJUNCTION FORMULAS 91 3 3 FINITENESS PROPERTY 93 3 4 MINIMAL EXTENSIONS 95 4 ANALYTIC D MODULES AND THE DE RHAM FUNCTOR 99 4 1 ANALYTIC D MODULES 99 4 2 SOLUTION COMPLEXES AND DE RHAM FUNCTORS 103 4 3 CAUCHY KOWALEVSKI KASHIWARA THEOREM 105 4 4 CAUCHY PROBLEMS AND MICRO SUPPORTS 107 4 5 CONSTRUCTIBLE SHEAVES 110 4 6 KASHIWARA S CONSTRUCTIBILITY THEOREM 113 4 7 ANALYTIC D MODULES ASSOCIATED TO ALGEBRAIC D MODULES 119 5 THEORY OF MEROMORPHIC CONNECTIONS 127 5 1 MEROMORPHIC CONNECTIONS IN THE ONE DIMENSIONAL CASE 127 5 1 1 SYSTEMS OF ODES AND MEROMORPHIC CONNECTIONS 127 5 1 2 MEROMORPHIC CONNECTIONS WITH REGULAR SINGULARITIES 130 5 1 3 REGULARITY OF D MODULES ON ALGEBRAIC CURVES 135 5 2 REGULAR MEROMORPHIC CONNECTIONS ON COMPLEX MANIFOLDS 139 5 2 1 MEROMORPHIC CONNECTIONS IN HIGHER DIMENSIONS 139 5 2 2 MEROMORPHIC CONNECTIONS WITH LOGARITHMIC POLES 143 5 2 3 DELIGNE S RIEMANN HILBERT CORRESPONDENCE 147 5 3 REGULAR INTEGRABLE CONNECTIONS ON ALGEBRAIC VARIETIES 153 6 REGULAR HOLONOMIC D MODULES 161 6 1 DE NITION AND MAIN THEOREMS 161 6 2 PROOF OF MAIN THEOREMS 163 7 RIEMANN HILBERT CORRESPONDENCE 171 7 1 COMMUTATIVITY WITH DE RHAM FUNCTORS 171 7 2 RIEMANN HILBERT CORRESPONDENCE 174 7 3 COMPARISON THEOREM 178 8 PERVERSE SHEAVES 181 8 1 THEORY OF PERVERSE SHEAVES 181 8 1 1 T STRUCTURES 181 8 1 2 PERVERSE SHEAVES 191 8 2 INTERSECTION COHOMOLOGY THEORY 202 8 2 1 INTRODUCTION 202 8 2 2 MINIMAL EXTENSIONS OF PERVERSE SHEAVES 203 8 3 HODGE MODULES 217 8 3 1 MOTIVATION 217 8 3 2 HODGE STRUCTURES AND THEIR VARIATIONS 217 8 3 3 HODGE MODULES 220 PART II REPRESENTATION THEORY 9 ALGEBRAIC GROUPS AND LIE ALGEBRAS 229 9 1 LIE ALGEBRAS AND THEIR ENVELOPING ALGEBRAS 229 9 2 SEMISIMPLE LIE ALGEBRAS 1 231 9 3 ROOT SYSTEMS 233 9 4 SEMISIMPLE LIE ALGEBRAS 2 236 9 5 FINITE DIMENSIONAL REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS 239 9 6 ALGEBRAIC GROUPS AND THEIR LIE ALGEBRAS 241 9 7 SEMISIMPLE ALGEBRAIC GROUPS 243 9 8 REPRESENTATIONS OF SEMISIMPLE ALGEBRAIC GROUPS 248 9 9 FLAG MANIFOLDS 250 9 10 EQUIVARIANT VECTOR BUNDLES 253 9 11 THE BOREL WEIL BOTT THEOREM 255 10 CONJUGACY CLASSES OF SEMISIMPLE LIE ALGEBRAS 259 10 1 THE THEORY OF INVARIANT POLYNOMIALS 259 10 2 CLASSI CATION OF CONJUGACY CLASSES 263 10 3 GEOMETRY OF CONJUGACY CLASSES 266 11 REPRESENTATIONS OF LIE ALGEBRAS AND D MODULES 271 11 1 UNIVERSAL ENVELOPING ALGEBRAS AND DIFFERENTIAL OPERATORS 271 11 2 RINGS OF TWISTED DIFFERENTIAL OPERATORS ON AG VARIETIES 272 11 3 PROOF OF THEOREM 11 2 2 276 11 4 PROOF OF THEOREMS 11 2 3 AND 11 2 4 280 11 5 EQUIVARIANT REPRESENTATIONS AND EQUIVARIANT D MODULES 283 11 6 CLASSI CATION OF EQUIVARIANT D MODULES 286 12 CHARACTER FORMULA OF HIGHEST WEIGHT MODULES 289 12 1 HIGHEST WEIGHT MODULES 289 12 2 KAZHDAN LUSZTIG CONJECTURE 295 12 3 D MODULES ASSOCIATED TO HIGHEST WEIGHT MODULES 299 13 HECKE ALGEBRAS AND HODGE MODULES 305 13 1 WEYL GROUPS AND D MODULES 305 13 2 HECKE ALGEBRAS AND HODGE MODULES 310 INTRODUCTION THE THEORY OF D MODULES PLAYS A KEY ROLE IN ALGEBRAIC ANALYSIS FOR THE PURPOSES OF THIS TEXT BY ALGEBRAIC ANALYSIS WE MEAN ANALYSIS USING ALGEBRAIC METHODS SUCH AS RING THEORY AND HOMOLOGICAL ALGEBRA IN ADDITION TO THE CONTRIBUTIONS BY FRENCH MATHEMATICIANS J BERNSTEIN AND OTHERS THIS AREA OF RESEARCH HAS BEEN EXTENSIVELY DEVELOPED SINCE THE 1960S BY JAPANESE MATHEMATICIANS NOTABLY IN THE IMPORTANT CONTRIBUTIONS OF M SATO T KAWAI AND M KASHIWARA OF THE KYOTO SCHOOL TO THIS DAY THERE CONTINUE TO BE OUTSTANDING RESULTS AND SIGNI CANT THEORIES COM ING FROM THE KYOTO SCHOOL INCLUDING SATO S HYPERFUNCTIONS MICROLOCAL ANALYSIS D MODULES AND THEIR APPLICATIONS TO REPRESENTATION THEORY AND MATHEMATICAL PHYSICS IN PARTICULAR THE THEORY OF REGULAR HOLONOMIC D MODULES AND THEIR SOLUTION COMPLEXES E G THE THEORY OF THE RIEMANN HILBERT CORRESPONDENCE WHICH GAVE A SOPHISTICATED ANSWER TO HILBERT S 21ST PROBLEM WAS A MOST IMPORTANT AND IN UENTIAL RESULT INDEED IT PROVIDED THE GERM FOR THE THEORY OF PERVERSE SHEAVES WHICH WAS A NATURAL DEVELOP MENT FROM INTERSECTION COHOMOLOGIES MOREOVER M SAITO USED THIS RESULT EFFECTIVELY TO CONSTRUCT HIS THEORY OF HODGE MODULES WHICH LARGELY EXTENDED THE SCOPE OF HODGE THEORY IN REPRESENTATION THEORY THIS RESULT OPENED TOTALLY NEW PERSPECTIVES SUCH AS THE RESOLUTION OF THE KAZHDAN LUSZTIG CONJECTURE AS STATED ABOVE IN ADDITION TO THE STRONG IMPACT ON ANALYSIS WHICH WAS THE INITIAL MAIN MOTIVATION THE THEORY OF ALGEBRAIC ANALYSIS ESPECIALLY THAT OF D MODULES CONTINUES TO PLAY A CENTRAL ROLE IN VARIOUS ELDS OF CONTEMPORARY MATHEMATICS IN FACT D MODULE THEORY IS A SOURCE FOR CREATING NEW RESEARCH AREAS FROM WHICH NEW THEORIES EMERGE THIS STRIKING FEATURE OF D MODULE THEORY HAS STIMULATED MATHEMATICIANS IN VARIOUS OTHER ELDS TO BECOME INTERESTED IN THE SUBJECT OUR AIM IS TO GIVE A COMPREHENSIVE INTRODUCTION TO D MODULES UNTIL RECENTLY IN ORDER TO REALLY LEARN IT WE HAD TO READ AND BECOME FAMILIAR WITH MANY ARTICLES WHICH TOOK LONG TIME AND CONSIDERABLE EFFORT HOWEVER AS WE MENTIONED IN THE PREFACE THANKS TO SOME TEXTBOOKS AND MONOGRAPHS THE THEORY HAS BECOME MUCH MORE ACCESSIBLE NOWADAYS ESPECIALLY FOR THOSE WHO HAVE SOME BASIC KNOWLEDGE OF COMPLEX ANALYSIS OR ALGEBRAIC GEOMETRY STILL TO UNDERSTAND AND APPRECIATE THE REAL SIGNI CANCE OF THE SUBJECT ON A DEEP LEVEL IT WOULD BE BETTER TO LEARN BOTH THE THEORY AND ITS TYPICAL APPLICATIONS