The Grobner fan of an A(n)-module

Let I be a non-zero left ideal of the Weyl algebra A(n) of order n over a f ield k and let L : R-2n --> R be a linear form defined by L(alpha, beta) = Sigma(i-l)(n) e(i)alpha(i) + Sigma(i=l)(n) f(i)beta(i). If e(i) + f(i) grea ter than or equal to 0, then L defines a filtration F.(L) on A(n). Let gr(L )(I) be the graded ideal associated with the filtration induced by F.(L) on I. Let finally U denote the set of all linear form L for which e(i) + f(i) greater than or equal to 0 for all 1 less than or equal to i less than or equal to n. The aim of this paper is to study, by using the theory of Grobn er bases, the stability of grL(I) when L varies in U, In a previous paper, we obtained finiteness results for some particular linear forms (used in or der to study the regularity of a D-module along a smooth hypersurface). Her e we generalize these results by adapting the theory of Grobner fall of Mor a-Robbiano to the D-module case. Our main tool is the homogenization techni que initiated in our previous paper, and recently clarified in a work by F. Castro-Jimenez and L, Narvaez-Macarro. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. Primary 35A27; secondary 13P10; 68Q40.