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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

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$47
Individual Members: US$28.20
Institutional Members: US$37.60
Order Code: MEMO/136/650
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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.

Table of Contents

  • 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
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