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We consider the question: what is the contragredient in terms ofL-homomorphisms? We conjecture that it corresponds to the Chevalleyautomorphism of the L-group, and prove this in the case of real groups. Theproof uses a characterization of the local Langlands correspondence over R. Wealso consider the related notion of Hermitian dual, in the case of GL(n,R).

Contragredient representations and characterizing the local Langlands correspondence