  We study holonomic modules for the rings of invariant differential operators
on affine commutative domains with finite Krull dimension with respect to
arbitrary actions of finite groups. We prove the Bernstein inequality for these
rings. Our main tool is the filter dimension introduced by Bavula. We extend
the results for the invariants of the Weyl algebra with respect to the
symplectic action of a finite group, for the rings of invariant differential
operators on quotient varieties, and invariants of certain generalized Weyl
algebras under the linear actions. We show that the filter dimension of all
above mentioned algebras equals 1.
