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 $\mathfrak{g}\subseteq\mathfrak{gl}_N$ 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.