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Given two isomorphic representations of a simple lie algebra over ℂ induced from two irreducible representations V and W of a parabolic subalgebra, what can be said about V and W? In this talk, we consider the case of the general linear and symplectic Lie algebras. We show that the dominant weights labelling V and W have to coincide up to a Dynkin diagram automorphism. In other terms, induction is injective. The proof relies on a generalised version of the Howe duality, which we will also explain. This is joint work with J. Guilhot and C. Lecouvey.
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