Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

Abstract

It is proved that the filter dimenion is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring D(X) of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand-Kirillov dimension GK (M ) ≥ n = GK (A) 2 for all nonzero finitely generated A-modules M . In fact, a more strong result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. As a direct consequence of this fact an affirmative answer is given to the question/conjecture posed by Ken Brown several years ago of whether an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H c for integral c: GK (M ) ≥ n = GK (Hc) 2 for all nonzero finitely generated H c -modules M .

Holonomic modules for rings of invariant differential operators

“…, where G a finite group of symplectic automorphisms of the Weyl algebra A n (k); • D(A n /G), k is algebraically closed and G is a finite group of linear automorphisms of the affine space A n k . We also extend the results of [10] for the invariants of generalized Weyl algebras under suitable actions of complex reflection groups of type G(m, p, r). Namely, we have Theorem 1.2.…”

“…It was shown in [10] that, given a finite Coxeter group W action on the Weyl algebra A n (k), the Bernstein inequality holds for A n (k) W : for every finitely generated A n (k) W -module M , GK(M ) ≥ n. We generalize this result and prove the Bernstein inequality for A n (k) G with linear action of an arbitrary finite group G. We note that our approach is different from the one in [10] (cf. Theorem 5.7 bellow).…”