Filtered Category OS and Applications.

Intro -- CONTENTS -- I. INTRODUCTION -- 1. INTRODUCTION -- 1.1. Summary -- 1.2. Filtered modules and filtered characters -- 1.3. Projective and self-dual modules -- 1.4. Overview of paper -- 1.5. Acknowledgements -- II. THE HECKE MODULE -- 2. THE SIMPLE AND VERMA BASES -- 2.1. Kazhdan-Lusztig polynomials -- 2.2. The Hecke module: four equivalent definitions -- 2.3. The Hecke module: proof of equivalences -- 2.4. Duality and the simple basis -- 2.5. The Hecke algebra action -- 3. THE PROJECTIVE AND SELF-DUAL BASES -- 3.1. The projective basis -- 3.2. Some elements of the Hecke algebra -- 3.3. An involution of M[sup(*)]sub(s)] -- 3.4. A new self-dual basis -- 4. THE SPECIALIZED HECKE MODULE -- 4.1. Definitions -- 4.2. Duality polynomials -- 4.3. A Jantzen sum formula -- III. THE FILTERED CATEGORY O[sub(s)] -- 5. THE CATEGORY O AND TRANSLATION FUNCTORS -- 5.1. The category and its Grothendieck group -- 5.2. Translation functors -- 6. THE FILTERED CATEGORY O[sub(s)] AND TRANSLATION FUNCTORS -- 6.1. Definition and basic properties of the filtered category -- 6.2. Some important filtrations -- 6.3. Translation functors on the filtered category -- IV. APPLICATIONS -- 7. RADICAL FILTRATIONS OF GENERALIZED VERMA MODULES AND PROJECTIVE MODULES -- 7.1. Radical filtrations of generalized Verma modules -- 7.2. Radical filtrations of projective indecomposable modules -- 7.3. More on radical filtrations -- 7.4. Rigidity of generalized Verma modules and projective modules -- 8. SELF-DUAL PROJECTIVE MODULES AND THEIR FILTERED CHARACTERS -- 8.1. Self-dual projective modules: review -- 8.2. Filtered self-duality of certain projective modules -- 8.3. Applications -- 9. SELF-DUAL MODULES WITH A VERMA FLAG AND THEIR FILTERED CHARACTERS -- 9.1. A class of self-dual modules with a Verma flag -- 9.2. A category of self-dual modules -- REFERENCES.

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