Adams, Vogan, and D. Prasad have given conjectural formulas for the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. We prove these conjectures for tempered representations of quasisplit real -groups and quasisplit -adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.
"Genericity and contragredience in the local Langlands correspondence." Algebra Number Theory 7 (10) 2447 - 2474, 2013. https://doi.org/10.2140/ant.2013.7.2447