FOURIER TRANSFORMS OF INVARIANT FUNCTIONS ON FINITE REDUCTIVE LIE ALGEBRAS

By V. KIUNK

CONTENTS INTRODUCTION CONNECTED REDUCTIVE GROUPS AND THEIR LIE ALGEBRAS NOTATION AND BACKGROUND H-VARIETIES AND ADJOINT ACTION OF H ON H REDUCTIVE GROUPS 2 1 10 ABOUT INTERSECTIONS OF LIE ALGEBRAS OF CLOSED SUBGROUPS OF G 2 1 16 FQ-STRUCTURES CHEVALLEY FORMULAS THE LIE ALGEBRA OF ZG EXISTENCE OF CHEVALLEY BASES ON G? EXISTENCE OF NON-DEGENERATE G-INVARIANT BILINEAR FORMS ON G CENTRALIZERS THE VARIETIES GUNI AND GNIL DELIGNE-LUSZTIG INDUCTION THE SPACE OF GF -INVARIANT FUNCTIONS ON GF DELIGNE-LUSZTIG INDUCTION: DE?NITION AND BASIC PROPERTIES DELIGNE-LUSZTIG INDUCTION: THE GROUP CASE DELIGNE-LUSZTIG INDUCTION: THE LIE ALGEBRA CASE 3 2 17 BASIC PROPERTIES OF RG L?P LOCAL SYSTEMS AND PERVERSE SHEAVES SIMPLE PERVERSE SHEAVES, INTERSECTION COHOMOLOGY COMPLEXES H-EQUIVARIANCE LOCALLY (ISO)TRIVIAL PRINCIPAL H-BUNDLES F-EQUIVARIANT SHEAVES AND COMPLEXES GEOMETRICAL INDUCTION ADMISSIBLE COMPLEXES AND ORBITAL PERVERSE SHEAVES ON G PARABOLIC INDUCTION OF EQUIVARIANT PERVERSE SHEAVES THE COMPLEXES INDG L?PK(?, E) 5 1 14 THE COMPLEXES INDG L?PK(?, E) ARE G-EQUIVARIANT PERVERSE SHEAVES 5 1 26 WHEN THE COMPLEXES INDG L?PK(?, E) ARE INTERSECTION COHOMOLOGY COMPLEXES 5 1 41 RESTRICTION OF INDG L?PK(?, E) TO G? WITH ? ?Z(G) 5 1 51 INTRODUCING FROBENIUS 5 1 56 ADMISSIBLE COMPLEXES (OR CHARACTER SHEAVES) ON G 5 1 72 ORBITAL PERVERSE SHEAVES: THE FUNDAMENTAL THEOREM DELIGNE-FOURIER TRANSFORMS AND ADMISSIBLE COMPLEXES ENDOMORPHISM ALGEBRA OF LUSZTIG COMPLEXES GEOMETRICAL INDUCTION: DE?NITION PRELIMINARIES 5 4 10 GEOMETRICAL INDUCTION DELIGNE-LUSZTIG INDUCTION AND GEOMETRICAL INDUCTION GENERALIZED GREEN FUNCTIONS THE CHARACTER FORMULA 5 5 11 GENERALIZED GREEN FUNCTIONS AND TWO-VARIABLE GREEN FUNCTIONS 5 5 14 GEOMETRICAL INDUCTION AND DELIGNE-LUSZTIG INDUCTION DELIGNE-LUSZTIG INDUCTION AND FOURIER TRANSFORMS FROBENIUS ACTION ON THE PARABOLIC INDUCTION OF CUSPIDAL ORBITAL PERVERSE SHEAVES THE FUNCTOR INDS×G S×L,P : ML(S × L) ?DB C(S × G) THE COMPLEXES INDS×G S×L,PK(Z × C, E) 6 1 15 THE COMPLEXES K1 AND K2 6 1 19 THE CHARACTER FORMULA 6 1 54 DELIGNE-LUSZTIG INDUCTION AND GEOMETRICAL INDUCTION ON THE CONJECTURE 3 2 30 REDUCTION OF 3 2 30 TO THE CASE OF NILPOTENTLY SUPPORTED CUSPIDAL FUNCTIONS THE MAIN RESULTS 6 2 20 LUSZTIG CONSTANTS: A FORMULA FOURIER TRANSFORMS OF THE CHARACTERISTIC FUNCTIONS OF THE ADJOINT ORBITS A DECOMPOSITION OF C(GF ) A GEOMETRIC ANALOGUE OF 3 2 24 FOURIER TRANSFORMS OF THE CHARACTERISTIC FUNCTIONS OF THE SEMI-SIMPLE ORBITS