Cover -- Half-title -- Title -- Copyright -- Table of Contents -- Preface -- Terminology -- PART I: Representing finite BN-pairs -- 1 Cuspidality in finite groups -- 1.1. Subquotients and associated restrictions -- 1.2. Cuspidality and induction -- 1.3. Morphisms and an invariance theorem -- 1.4. Endomorphism algebras of induced cuspidal modules -- 1.5. Self-injective endomorphism rings and an equivalence of categories -- 1.6. Structure of induced cuspidal modules and series -- Notes -- 2 Finite BN-pairs -- 2.1. Coxeter groups and root systems -- 2.2. BN-pairs -- 2.3. Root subgroups -- 2.4. Levi decompositions -- 2.5. Other properties of split BN-pairs -- Notes -- 3 Modular Hecke algebras for finite BN-pairs -- 3.1. Hecke algebras in transversal characteristics -- 3.2. Quotient root system and a presentation of the Hecke algebra -- Notes -- 4 The modular duality functor and derived category -- 4.1. Homology -- 4.1.1. Complexes and associated categories -- 4.1.2. Simplicial schemes -- 4.1.3. Coefficient systems -- 4.1.4. Associated homology complexes -- 4.2. Fixed point coefficient system and cuspidality -- 4.3. The case of finite BN-pairs -- 4.4. Duality functor as a derived equivalence -- 4.5. A theorem of Curtis type -- Notes -- 5 Local methods for the transversal characteristics -- 5.1. Local methods and two main theorems of Brauer's -- 5.2. A model: blocks of symmetric groups -- 5.3. Principal series and the principal block -- 5.4. Hecke algebras and decomposition matrices -- 5.5. A proof of Brauer's third Main Theorem -- Notes -- 6 Simple modules in the natural characteristic -- 6.1. Modular Hecke algebra associated with a Sylow p-subgroup -- 6.2. Some modules in characteristic p -- 6.3. Alperin's weight conjecture in characteristic p -- 6.4. The p-blocks -- Notes.

PART II: Deligne-Lusztig varieties, rational series, and Morita equivalences -- 7 Finite reductive groups and Deligne-Lusztig varieties -- 7.1. Reductive groups and Lang's theorem -- 7.2. Varieties defined by the Lang map -- 7.3. Deligne-Lusztig varieties -- 7.4. Deligne-Lusztig varieties are quasi-affine -- Notes -- 8 Characters of finite reductive groups -- 8.1. Reductive groups, isogenies -- 8.2. Some exact sequences and groups in duality -- 8.3. Twisted induction -- 8.4. Lusztig's series -- Notes -- 9 Blocks of finite reductive groups and rational series -- 9.1. Blocks and characters -- 9.2. Blocks and rational series -- 9.3. Morita equivalence and ordinary characters -- Notes -- 10 Jordan decomposition as a Morita equivalence: the main reductions -- 10.1. The condition… -- 10.2. A first reduction -- 10.3. More notation: smooth compactifications -- 10.4. Ramification and generation -- 10.5. A second reduction -- Notes -- 11 Jordan decomposition as a Morita equivalence: sheaves -- 11.1. Ramification 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 -- Notes -- 12 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 -- Notes -- PART III: Unipotent characters and unipotent blocks -- 13 Levi subgroups and polynomial orders -- 13.1. Polynomial orders of F-stable tori -- 13.2. Good primes -- 13.3. Centralizers of l-subgroups and some Levi subgroups -- Notes -- 14 Unipotent characters as a basic set -- 14.1. Dual conjugacy classes for l-elements -- 14.2. Basic sets in the case of connected center -- Notes.

15 Jordan decomposition of characters -- 15.1. From non-connected center to connected center and dual morphism -- 15.2. Jordan decomposition of characters -- Notes -- 16 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 -- 16.8. Last computation -- Notes -- 17 Standard isomorphisms for unipotent blocks -- 17.1. The set of unipotent blocks -- 17.2. l-series and non-connected center -- 17.3. A ring isomorphism -- Note -- PART IV: Decomposition numbers and q-Schur algebras -- 18 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 -- Notes -- 19 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 -- Notes -- 20 Decomposition numbers and Fei-Schur algebras: linear primes -- 20.1. Finite classical groups and linear primes -- 20.2. Hecke algebras -- 20.3. Type BC -- 20.4. Type D -- Note -- PART V: Unipotent blocks and twisted induction -- 21 Local methods -- twisted induction for blocks -- 21.1. "Connected" subpairs in finite reductive groups.

21.2. Twisted induction for blocks -- 21.3. A bad prime -- Notes -- 22 Unipotent blocks and generalized Harish-Chandra theory -- 22.1. Local subgroups in finite reductive groups, l-elements and tori -- 22.2. The theorem -- 22.3. Self-centralizing subpairs -- 22.4. The defect groups -- Notes -- 23 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 -- Notes -- Appendices -- Appendix 1: Derived categories and derived functors -- A1.1. Abelian categories -- A1.2. Complexes and standard constructions -- A1.3. The mapping cone -- A1.4. Homology -- A1.5. The homotopic category -- A1.6. Derived categories -- A1.7. Cones and distinguished triangles -- A1.8. Derived functors -- A1.9. Composition of derived functors -- A1.10. Exact sequences of functors -- A1.11. Bi-functors -- A1.12. Module categories -- A1.13. Sheaves on topological spaces -- A1.14. Locally constant sheaves and the fundamental group -- A1.15. Derived operations on sheaves -- Notes -- Appendix 2: Varieties and schemes -- A2.1. Affine F-varieties -- A2.2. Locally ringed spaces and F-varieties -- A2.3. Tangent sheaf, smoothness -- A2.4. Linear algebraic groups and reductive groups -- A2.5. Rational structures on affine varieties -- A2.6. Morphisms and quotients -- A2.7. Schemes -- A2.8. Coherent sheaves -- A2.9. Vector bundles -- A2.10. A criterion of quasi-affinity -- Notes -- Appendix 3: Etale cohomology -- A3.1. The étale topology -- A3.2. Sheaves for the étale topology -- A3.3. Basic operations on sheaves -- A3.4. Homology and derived functors -- A3.5. Base change for a proper morphism -- A3.6. Homology and direct images with compact support -- A3.7. Finiteness of cohomology -- A3.8. Coefficients -- A3.9. The "open-closed" situation.

A3.10. Higher direct images and stalks -- A3.11. Projection and Künneth formulae -- A3.12. Poincaré-Verdier duality and twisted inverse images -- A3.13. Purity -- A3.14. Finite group actions and constant sheaves -- A3.15. Finite group actions and projectivity -- A3.16. Locally constant sheaves and the fundamental group -- A3.17. Tame ramification along a divisor with normal crossings -- A3.18. Tame ramification and direct images -- Notes -- References -- Index.