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We completely determine the minimal polynomial of an arbitrary simple highestweight 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 idealsand algebraic properties of Howe duality correspondence.

Minimal polynomials of simple highest weight modules over classical Lie algebras