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Invariants under Tori of Rings of Differential Operators and Related Topics
Ian M. Musson, University of Wisconsin, Milwaukee, WI, and Michel Van den Bergh, Free University of Brussels, Belgium
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Memoirs of the American Mathematical Society
1998; 85 pp; softcover
Volume: 136
ISBN-10: 0-8218-0885-0
ISBN-13: 978-0-8218-0885-6
List Price: US$44 Individual Members: US$26.40
Institutional Members: US\$35.20
Order Code: MEMO/136/650

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.

Graduate students and research mathematicians working in rings of differential operators; algebraic geometers and others interested in toric varieties.

• Introduction
• Notations and conventions
• A certain class of rings
• Some constructions
• The algebras introduced by S. P. Smith
• The Weyl algebras
• Rings of differential operators for torus invariants
• Dimension theory for $$B^\chi$$
• Finite global dimension
• Finite dimensional representations
• An example
• References