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 paper we show that this is indeed the case when G is a torus and X = k r × (k * ) s . We give a precise description of the primitive ideals in D(X) G and we study in detail the ring theoretical and homological properties of the minimal primitive quotients of D(X) G . The latter

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... are of the form D(X) G /(g − χ(g)) where g = Lie(G), χ ∈ g * and g − χ(g) is the set of all v − χ(v) with v ∈ g. They occur as rings of twisted differential operators on toric varieties. As a side result we prove that if G is a torus acting rationally on a smooth affine variety then D(X/ /G) is a simple ring.