# Invariants under Tori of Rings of Differential Operators and Related Topics

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*Ian M. Musson; Michel Van den Bergh*

If \(G\) is a reductive algebraic group acting rationally on a smooth affine variety \(X\), then it is generally believed that \(D(X)^G\) has properties very similar to those of enveloping algebras of semisimple Lie algebras. In this book, the authors show that this is indeed the case when \(G\) is a torus and \(X=k^r\times (k^*)^s\). They give a precise description of the primitive ideals in \(D(X)^G\) and study in detail the ring theoretical and homological properties of the minimal primitive quotients of \(D(X)^G\). The latter are of the form \(B^x=D(X)^G/({\mathfrak g}-\chi({\mathfrak g}))\) where \({\mathfrak g}= \mathrm{Lie}(G)\), \(\chi\in {\mathfrak g}^\ast\) and \({\mathfrak g}-\chi({\mathfrak g})\) is the set of all \(v-\chi(v)\) with \(v\in {\mathfrak g}\). They occur as rings of twisted differential operators on toric varieties. It is also proven that if \(G\) is a torus acting rationally on a smooth affine variety, then \(D(X/\!/G)\) is a simple ring.

#### Table of Contents

# Table of Contents

## Invariants under Tori of Rings of Differential Operators and Related Topics

- Contents vii8 free
- 1. Introduction 110 free
- 2. Notations and conventions 413 free
- 3. A certain class of rings 413
- 4. Some constructions 1827
- 5. The algebras introduced 2332
- 6. The Weyl algebras 3140
- 7. Rings of differential operators for torus invariants 3342
- 8. Dimension theory for B[sup(x)] 4655
- 9. Finite global dimension 5564
- 10. Finite dimensional representations 6978
- 11. An example 7584
- References 8392