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. In this paper we show that this is indeed the case when G is a torus and X = k(r) x (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 are of the form D(X)(G)/(g - chi(g)) where g = Lie(G), chi is an element of g* and g - chi(g) is the set of all upsilon - chi(upsilon) with upsilon is an element of 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.