Invariants under tori of rings of differential operators and related topics Book

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 $X=k^rtimes (k^*)^s$. 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 ${mathfrak g}= mathrm{Lie}(G)$, $chiin {mathfrak g}^ast$ and ${mathfrak g}-chi({mathfrak g})$ is the set of all $v-chi(v)$ with $vin {mathfrak g}$. 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.