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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