Mickelsson algebras and the inverse Shapovalov form

Let $\mathcal{A}$ be an associative algebra containing the classical or quantum universal enveloping algebra $U$ of a semisimple complex Lie algebra. Let $\mathcal{J}\subset \mathcal{A}$ designate the left ideal generated by positive root vectors in $U$. The reduction algebra of the pair $(\mathcal{A},\mathcal{J})$ is constructed via the inverse Shapovalov form of $U$.