Invariants Under Tori of Rings of Differential Operators and Related Topics

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. This book shows that this is indeed the case when $G$ is a torus and $X=k^r/times (k^*)^s$.

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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 ${/mathfrak g}=/textnormal{Lie}(G)$, $/chi/in {/mathfrak g}^/ast$ 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 $D(X[LAMBDA]!/G)$ is a simple ring.