ABSTRACT We consider rings of differential operators over the classical rings of invariants, in the sense of Weyl [We]. Thus, let Xk be one of the following varieties: (CASE A) all complex p x q matrices of rank h (CASE B) all symmetric n x n matrices of rank h (CASE C) all antisymmetric n x n matrices of rank 2k. We prove that the ring of differential operators V(Xk) = V(0(X k)) defined on the ring of regular functions O(Xk) is a simple, finitely generated, Noetherian domain. Assume further that Xk is singular (which is the only interesting case). Then the result is proved by showing that 'D(Xk) is a factor ring of an enveloping algebra U(Q) . Here 9 = gl(p + g), sp(2n) and so(2n) in the Cases A, B and C, respectively. Finally, let SO(k) act in the natural way on the ring C[X] of complex polynomials in kn variables. Then we prove that V(C[X]so^) has a similarly pleasant structure and, at least for k n , is a finitely generated U(sp(2n)) -module. Key words and phrases Rings of differential operators, classical rings of invariants, semi-simple Lie algebras, primitive factor rings, simple rings, Noetherian rings. IV

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