CONTENTS Abstract iv Introduction 1 Index of Notation 8 Chapter I. Reductive dual pairs and the Howe Correspondence 10 LI. Reductive dual pairs 10 1.2. Formulae for the metaplectic representation 13 1.3. Preliminary results on the structure of S(U)G and Ker(ip) 15 Chapter II. Classical reductive dual pairs : explicit calculations 18 11.1. Introduction 18 11.2. Description of Case A : GL(p + q) x GL(k) 19 11.3. Description of Case B : Sp(2n) x 0(k) 24 11.4. Description of Case C : 0(2n) x Sp(2k) 27 11.5. Comments and notation 30 11.6. The associated variety of J(k) = ker(rf) 34 Chapter III. Differential operators on classical rings of invariants 38 111.1. Reduction of the main theorem 38 111.2. Dimensions of associated varieties 45 111.3. Twisted differential operators 61 Chapter IV. The maximality of J(k) and the simplicity of V{X\^) 65 IV. 1. Introduction and consequences of the maximality of J{k) 65 IV.2. Outline of the proof of the maximality of J(k) 72 IV.3. The maximality of J(k) in Case A 76 IV.4. The maximality of J(k) in Case B 79 IV.5. The maximality of J(k) in Case C 84 Chapter V. Differential operators on the ring of SO(k)-invariants 86 V.l. Introduction and background 86 V.2. Differential operators on 0(X)S0 k) for k ^ n 90 V.3. Differential operators on 0(X)S0{k) when k = n 93 V.4. On the identity 50(2 ) = GL(l) 102 V.5. The structure of V(0{X)so^) as a U(sp(2n))-module 103 Appendix. Gabber's Lemma 107 References 114 iii

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