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
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