Whittaker Normalization of <i>p</i> -Adic ABV-Packets and Vogan’s Conjecture for Tempered Representations

Abstract We show that ABV-packets for $p$-adic groups do not depend on the choice of a Whittaker datum, but the function from the ABV-packet to representations of the appropriate equivariant fundamental group does, and we find this dependence exactly. We give a geometric characterization of generic $L$-parameters and tempered $L$-parameters. We state a genericity conjecture for ABV-packets and prove this conjecture for quasi-split classical groups and their pure inner forms. Motivated by this, we study ABV-packets for open parameters and prove that they are $L$-packets, and further that the function from the packet to the equivariant fundamental group given by normalized vanishing cycles coincides with the one given by the Langlands correspondence. We conclude with Vogan’s conjecture on A-packets for tempered representations: ABV-packets for tempered parameters are Arthur packets and the function from the packet to the equivariant fundamental group given by normalized vanishing cycles coincides with the one given by Arthur.