Minimal polynomials of simple highest weight modules over classical Lie algebras

We completely determine the minimal polynomial of an arbitrary simple highest weight module $L(\lambda )$ over a complex classical Lie algebra $\mathfrak{g}\subseteq {\mathfrak{g}\mathfrak{l}}_N$ relative to its defining module $\pi ={\mathbb{C}}^N$. These results are applied to ordering on primitive ideals and algebraic properties of Howe duality correspondence.