INTRODUCTION 0.1. Given a commutative C -algebra R, the ring V(R) of C-linear differential operators on R is defined, inductively, as follows. Let T0(R) = Homn(R,R) = R. For m 0 define the set of differential operators of order m to be Vm(R) = {6e Endc(R) : [9, a] G Vm-i(R) for all a R}. oo Then T(R) = (J Vm(R), with multiplication defined by composition of functions. In m=0 order to avoid confusion with multiplication inside V{R), the action of 9 E 'D(R) as a differential operator on r £ R will always be written 9 * r . Basic facts about P(-R) can be found, for example, in [Sw] or [MR, Chapter XV]. Given a quasi-affine algebraic variety y, with regular functions 0{y), write T(y) for T(0(y)). One aspect of T(y) is of particular relevance here. If y is affine, non- singular and irreducible, it is well known that T( y) has a particularly pleasant structure being, in particular, a simple Noetherian domain that is finitely generated as a C -algebra (see, for example [MR, Chapter XV, §§1.20, 3.7 and 5.6] or [SmSt, §1.4]). In contrast, when y has singularities, V(y) need not be pleasant. For example, if y is the cubic cone x\ + x\ + x\ 0 in complex 3-space, then T{y) is neither simple nor Noetherian nor finitely generated, and even has an infinite ascending chain of two-sided ideals (see [BGG]). 0.2. The aim of this paper is to study the rings of differential operators on classical rings of invariants. Indeed, even though the varieties will often be singular, we will prove that the corresponding rings of differential operators will always be simple Noetherian domains, generated by the "obvious" differential operators of order 2. Following Weyl [We] (but see also [DP]) we will consider 3 main classes of rings of invariants. To describe them, let Mp^q(C) denote the space of p x q complex matrices, and fix k 1. (CASE A) Given p q 1, let G' = GL{k) = GL(k,C) act on X = MPik(C) x M M ( C ) by g-i^rj) = ((9'\gv) for g 6 GL(k) and (£,77) X. Then GL{k) acts on 0{X) and the ring of invariants 0(X)GL^ is described by the two Fundamental Theorems of Invariant Theory (see [DP] or (II, Theorem 2.3)). For the Received by the editors February 8, 1988 1
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