Contragredient representations and characterizing the local Langlands correspondence

It is surprising that the following question has not been addressed in the literature: what is the contragredient in terms of Langlands parameters? Thus suppose G is a connected, reductive algebraic group defined over a local field F , and G(F ) is its F -points. According to the local Langlands conjecture, associated to an admissible homomorphism φ from the WeilDeligne group of F into the L-group of G(F ) is an L-packet Π(φ), a finite set of (equivalence classes of) irreducible admissible representations of G(F ). (If F is archimedean, “equivalence” means “infinitesimal equivalence.” If F is non-archimedean, it means “equivalence of smooth vectors.”) Conjecturally these L-packets partition the admissible dual. So suppose π is an irreducible admissible representation, and π ∈ Π(φ). Let π∗ be the contragredient, or dual, of π (see (7.1)). The question is: what is the homomorphism φ∗ such that π∗ ∈ Π(φ∗)? We also consider the related question of describing the Hermitian dual in terms of Langlands parameters. Both of the questions come down to a characterization of the