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We prove the recent conjectures of Adams-Vogan and D. Prasad on the behaviorof the local Langlands correspondence with respect to taking the contragredientof a representation. The proof holds for tempered representations ofquasi-split real K-groups and quasi-split p-adic classical groups (in the senseof Arthur). We also prove a formula for the behavior of the local Langlandscorrespondence for these groups with respect to changes of the Whittaker data.

Genericity and contragredience in the local Langlands correspondence