Non-semisimple Macdonald Polynomials I

0. Introduction 3 0.1. Reduced decompositions 4 0.2. Affine exponents 7 0.3. Non–semisimple polynomials 12 0.4. Expected applications 17 1. Affine Weyl groups 21 1.1. Affine roots 22 1.2. The length on Ŵ 24 1.3. Reduction modulo W 29 1.4. Partial ordering on P 34 2. Reduced decompositions 37 2.1. Lambda–sequences 37 2.2. Coxeter transformations 39 2.3. Theorem about triples 42 2.4. Discussion 44 3. Induction process 47 3.1. Subsystems B3, C3 47 3.2. Admissible triples 49 3.3. Special cases 51 3.4. Uniform construction 54 3.5. The case mij > 1 56 3.6. Concluding the induction 59 4. Right Bruhat ordering 62 4.1. Basic properties 62 4.2. Using Coxeter transforms 64 4.3. Bruhat ordering on W̃ 0 69 5. Double Hecke algebras 71 5.1. Main Definition 72 5.2. Automorphisms 73 5.3. Intertwining operators 74 5.4. Intertwiners and Ti 75