Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters

Abstract: In this paper we prove the forward direction of the conjecture of Gross-Prasad that an L-packet $\Pi_\phi(G)$ contains a generic representation if and only if $L(s, \phi, \Ad)$ is regular at $s=1$, by assuming the local Langlands correspondence and the $p$-adic Kazhdan-Lusztig hypothesis. We then prove an analogous statement for ABV-packets, which together with Vogan's conjecture on ABV-packets implies that if Arthur's conjectures for $G$ are known, then one direction of Shahidi's enhanced genericity conjecture holds: If an Arthur packet $\Pi_\psi(G)$ contains a generic representation, then $\phi_\psi$ is tempered. In the case where the infinitesimal parameters in question are all unramified, we obtain converses to the above statements. We also offer some speculation about the relationship between Arthur type representations, and singularities in varieties of Langlands parameters defined by Vogan. Finally, we recover some facts about central characters using the results above.