REPRESENTATION THEORY OF FINITE REDUCTIVE GROUPS

By STEVEN THOMPSON

CONTENTS PART I REPRESENTING FINITE BN-PAIRS CUSPIDALITY IN FINITE GROUPS SUBQUOTIENTS AND ASSOCIATED RESTRICTIONS CUSPIDALITY AND INDUCTION MORPHISMS AND AN INVARIANCE THEOREM ENDOMORPHISM ALGEBRAS OF INDUCED CUSPIDAL MODULES SELF-INJECTIVE ENDOMORPHISM RINGS AND AN EQUIVALENCE OF CATEGORIES STRUCTURE OF INDUCED CUSPIDAL MODULES AND SERIES FINITE BN-PAIRS COXETER GROUPS AND ROOT SYSTEMS BN-PAIRS ROOT SUBGROUPS LEVI DECOMPOSITIONS OTHER PROPERTIES OF SPLIT BN-PAIRS MODULAR HECKE ALGEBRAS FOR ?NITE BN-PAIRS HECKE ALGEBRAS IN TRANSVERSAL CHARACTERISTICS QUOTIENT ROOT SYSTEM AND A PRESENTATION OF THE HECKE ALGEBRA THE MODULAR DUALITY FUNCTOR AND DERIVED CATEGORY HOMOLOGY FIXED POINT COEF?CIENT SYSTEM AND CUSPIDALITY THE CASE OF ?NITE BN-PAIRS DUALITY FUNCTOR AS A DERIVED EQUIVALENCE A THEOREM OF CURTIS TYPE LOCAL METHODS FOR THE TRANSVERSAL CHARACTERISTICS LOCAL METHODS AND TWO MAIN THEOREMS OF BRAUER’S A MODEL: BLOCKS OF SYMMETRIC GROUPS PRINCIPAL SERIES AND THE PRINCIPAL BLOCK HECKE ALGEBRAS AND DECOMPOSITION MATRICES A PROOF OF BRAUER’S THIRD MAIN THEOREM SIMPLE MODULES IN THE NATURAL CHARACTERISTIC MODULAR HECKE ALGEBRA ASSOCIATED WITH A SYLOW P-SUBGROUP SOME MODULES IN CHARACTERISTIC P ALPERIN’S WEIGHT CONJECTURE IN CHARACTERISTIC P THE P-BLOCKS PART II DELIGNE–LUSZTIG VARIETIES, RATIONAL SERIES, AND MORITA EQUIVALENCES FINITE REDUCTIVE GROUPS AND DELIGNE–LUSZTIG VARIETIES REDUCTIVE GROUPS AND LANG’S THEOREM VARIETIES DE?NED BY THE LANG MAP DELIGNE–LUSZTIG VARIETIES DELIGNE–LUSZTIG VARIETIES ARE QUASI-AF?NE CHARACTERS OF FINITE REDUCTIVE GROUPS REDUCTIVE GROUPS, ISOGENIES SOME EXACT SEQUENCES AND GROUPS IN DUALITY TWISTED INDUCTION LUSZTIG’S SERIES BLOCKS OF ?NITE REDUCTIVE GROUPS AND RATIONAL SERIES BLOCKS AND CHARACTERS BLOCKS AND RATIONAL SERIES MORITA EQUIVALENCE AND ORDINARY CHARACTERS JORDAN DECOMPOSITION AS A MORITA EQUIVALENCE: THE MAIN REDUCTIONS 10 1 THE CONDITION I?R J?F = 10 2 A ?RST REDUCTION 10 3 MORE NOTATION: SMOOTH COMPACTI?CATIONS 10 4 RAMI?CATION AND GENERATION 10 5 A SECOND REDUCTION JORDAN DECOMPOSITION AS A MORITA EQUIVALENCE: SHEAVES 11 1 RAMI?CATION IN DELIGNE–LUSZTIG VARIETIES 11 2 COROOT LATTICES ASSOCIATED WITH INTERVALS 11 3 DELIGNE–LUSZTIG VARIETIES ASSOCIATED WITH INTERVALS 11 4 APPLICATION: SOME MAPPING CONES JORDAN DECOMPOSITION AS A MORITA EQUIVALENCE: MODULES 12 1 GENERATING PERFECT COMPLEXES 12 2 THE CASE OF MODULES INDUCED BY DELIGNE–LUSZTIG VARIETIES 12 3 VARIETIES OF MINIMAL DIMENSION INDUCING A SIMPLE MODULE 12 4 DISJUNCTION OF SERIES PART III UNIPOTENT CHARACTERS AND UNIPOTENT BLOCKS LEVI SUBGROUPS AND POLYNOMIAL ORDERS 13 1 POLYNOMIAL ORDERS OF F-STABLE TORI 13 2 GOOD PRIMES 13 3 CENTRALIZERS OF ?-SUBGROUPS AND SOME LEVI SUBGROUPS UNIPOTENT CHARACTERS AS A BASIC SET 14 1 DUAL CONJUGACY CLASSES FOR ?-ELEMENTS 14 2 BASIC SETS IN THE CASE OF CONNECTED CENTER JORDAN DECOMPOSITION OF CHARACTERS 15 1 FROM NON-CONNECTED CENTER TO CONNECTED CENTER AND DUAL MORPHISM 15 2 JORDAN DECOMPOSITION OF CHARACTERS ON CONJUGACY CLASSES IN TYPE D 16 1 NOTATION; SOME POWER SERIES 16 2 ORTHOGONAL GROUPS 16 3 SPECIAL ORTHOGONAL GROUPS AND THEIR DERIVED SUBGROUP; CLIFFORD GROUPS 16 4 SPIN2N(F) 16 5 NON-SEMI-SIMPLE GROUPS, CONFORMAL GROUPS 16 6 GROUP WITH CONNECTED CENTER AND DERIVED GROUP SPIN2N (F); CONJUGACY CLASSES 16 7 GROUP WITH CONNECTED CENTER AND DERIVED GROUP SPIN2N (F); JORDAN DECOMPOSITION OF CHARACTERS STANDARD ISOMORPHISMS FOR UNIPOTENT BLOCKS 17 1 THE SET OF UNIPOTENT BLOCKS 17 2 ?-SERIES AND NON-CONNECTED CENTER 17 3 A RING ISOMORPHISM PART IV DECOMPOSITION NUMBERS AND Q-SCHUR ALGEBRAS SOME INTEGRAL HECKE ALGEBRAS 18 1 HECKE ALGEBRAS AND SIGN IDEALS 18 2 HECKE ALGEBRAS OF TYPE A 18 3 HECKE ALGEBRAS OF TYPE BC, HOEFSMIT’S MATRICES AND JUCYS–MURPHY ELEMENTS 18 4 HECKE ALGEBRAS OF TYPE BC: SOME COMPUTATIONS 18 5 HECKE ALGEBRAS OF TYPE BC: A MORITA EQUIVALENCE 18 6 CYCLIC CLIFFORD THEORY AND DECOMPOSITION NUMBERS DECOMPOSITION NUMBERS AND Q-SCHUR ALGEBRAS: GENERAL LINEAR GROUPS 19 1 HOM FUNCTORS AND DECOMPOSITION NUMBERS 19 2 CUSPIDAL SIMPLE MODULES AND GELFAND–GRAEV LATTICES 19 3 SIMPLE MODULES AND DECOMPOSITION MATRICES FOR UNIPOTENT BLOCKS 19 4 MODULAR HARISH-CHANDRA SERIES DECOMPOSITION NUMBERS AND Q-SCHUR ALGEBRAS: LINEAR PRIMES 20 1 FINITE CLASSICAL GROUPS AND LINEAR PRIMES 20 2 HECKE ALGEBRAS 20 3 TYPE BC 20 4 TYPE D PART V UNIPOTENT BLOCKS AND TWISTED INDUCTION LOCAL METHODS; TWISTED INDUCTION FOR BLOCKS 21 1 “CONNECTED” SUBPAIRS IN ?NITE REDUCTIVE GROUPS 21 2 TWISTED INDUCTION FOR BLOCKS 21 3 A BAD PRIME UNIPOTENT BLOCKS AND GENERALIZED HARISH-CHANDRA THEORY 22 1 LOCAL SUBGROUPS IN ?NITE REDUCTIVE GROUPS, ?-ELEMENTS AND TORI 22 2 THE THEOREM 22 3 SELF-CENTRALIZING SUBPAIRS 22 4 THE DEFECT GROUPS LOCAL STRUCTURE AND RING STRUCTURE OF UNIPOTENT BLOCKS 23 1 NON-UNIPOTENT CHARACTERS IN UNIPOTENT BLOCKS 23 2 CONTROL SUBGROUPS 23 3 (Q – 1)-BLOCKS AND ABELIAN DEFECT CONJECTURE