Rings of Differential Operators on Classical Rings of Invariants.

Intro -- Contents -- Abstract -- Introduction -- Index of Notation -- Chapter I. Reductive dual pairs and the Howe Correspondence -- I.1. Reductive dual pairs -- I.2. Formulae for the metaplectic representation -- I.3. Preliminary results on the structure of S(U)[sup(G')] and Ker(ψ) -- Chapter II. Classical reductive dual pairs: explicit calculations -- II.1. Introduction -- II.2. Description of Case A: GL(p+q) x GL(k) -- II.3. Description of Case B: Sp(2n) x O(k) -- II.4. Description of Case C: O(2n) x Sp(2k) -- II.5. Comments and notation -- II.6. The associated variety of J(k) = ker(ψ) -- Chapter III. Differential operators on classical rings of invariants -- III.1. Reduction of the main theorem -- III.2. Dimensions of associated varieties -- III.3. Twisted differential operators -- Chapter IV. The maximality of J(k) and the simplicity of D(Xk) -- IV.1. Introduction and consequences of the maximality of J{k) -- IV.2. Outline of the proof of the maximality of J(k) -- IV.3. The maximality of J(k) in Case A -- IV.4. The maximality of J(k) in Case B -- IV.5. The maximality of J(k) in Case C -- Chapter V. Differential operators on the ring of SO(k)-invariants -- V.1. Introduction and background -- V.2. Differential operators on O(X)[sup(SO(k))] for k ≠ n -- V.3. Differential operators on O(X)[sup(SO(k))] when k = n -- V.4. On the identity SO(2) = GL(1) -- V.5. The structure of D(O(X)[sup(SO(k))]) as a U(sp(2n))-module -- Appendix -- References.

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