Differential Operators and Weighted Isobaric Polynomials

We characterize those sequences of weighted isobaric polynomials as defined in math.CO/0106213 which belong to the kernel of the linear operator $D_{11} - \sum_{j=1}^k a_j t_j D_{2j} - mD_2$, and we characterize those linear operators of this form in terms of the coefficients $a_j$ which have a non-zero kernel.