The Contragredient
نویسندگان
چکیده
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 a homomorphism φ from the Weil-Deligne group of F into the L-group of G(F ) is an L-packet Π(φ), a finite set of irreducible admissible representations of G(F ). Conjecturally these sets partition the admissible dual. So suppose π is an irreducible admissible representation, and π ∈ Π(φ). Let π be the contragredient of π. 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. Let G be the complex dual group of G. The Chevalley involution C of G satisfies C(h) = h, for all h in some Cartan subgroup of G. The
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