Contents Introduction 1 1 Quantum linear groups and polynomial induction 9 1.1 Symmetric groups and Hecke algebras 9 1.2 The g-Schur algebra 11 1.3 Tensor products and Levi subalgebras 14 1.4 Polynomial induction 19 1.5 Schur algebra induction 25 2 Classical results on GLn 29 2.1 Conjugacy classes and Levi subgroups 29 2.2 Harish-Chandra induction and restriction 31 2.3 Characters and Deligne-Lusztig operators 34 2.4 Cuspidal representations and blocks 37 2.5 Howlett-Lehrer theory and the Gelfand-Graev representation 41 3 Connecting GLn with quantum linear groups 47 3.1 Schur functors 47 3.2 The cuspidal algebra 50 3.3 'Symmetric' and 'exterior' powers 54 3.4 Endomorphism algebras 58 3.5 Standard modules 63 4 Further connections and applications 67 4.1 Base change 67 4.2 Connecting Harish-Chandra induction with tensor products 71 4.3 p-Singular classes 75 4.4 Blocks and decomposition numbers 79 4.5 The Ringel dual of the cuspidal algebra 83 5 The affine general linear group 87 5.1 Levels and the branching rule from AGLn to GLn 87 5.2 Affine induction operators 93 5.3 The affine cuspidal algebra 98 5.4 The branching rule from GLn to AGLn-\ 101 5.5 A dimension formula for irreducibles 105 Bibliography 109 vii

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