Differential Operators on a Cubic Cone

Consider in the space C with the coordinates x{ , x2, x3 the surface X defined by the equation + + = 0. We prove the following theorem: T H E O R E M 1. Let D{X) be the ring of regular differential operators on X, and Da the ring of germs at the point 0 of analytic operators on X. Then 1°. the rings D(X) and Da are not Noetherian; 2°. for any natural number k the rings D{X) and Da are not generated by the subspaces Dk (Dak, respectively) of operators of order not exceeding k. In particular, the rings D{X) and Da are not finitely generated. Theorem 1 answers questions raised in Malgrange's survey article [ 1 ] . The ring D(X) has an interesting structure (see Proposition 1). We denote by E(X) the ring of regular functions on X{E(X) = C[x,, x2, x3]/[x + + x 3 3]) and by D(X) the ring of regular differential operators on X. By Dk we denote the space of operators of order not exceeding k. Setting ax{f){x) = /(λχ) and b\(3))(f) ~ aK(DaX-x(f)) for λ e C* we define an action of the group C* oo in the spaces E{X) and D(X). It is clear that E{X) = © E(X), where E{X) is the finite-dimensional space of homogenous functions of degree i on X. We call an operator 3 e D(X) homogenous of degree i (i e Z) if b%{3) = V-Sb for all λ e C* (equivalent definition: 2b {E {X)) C E(X) for all n). We denote by D' the space of all such operators and set D = D' η Dk.