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

Abstract

Let $G$ be a reductive $p$-adic group for which the local Langlands correspondence is known, and $\lambda$ an infinitesimal parameter of $G$. In this paper, we prove that if the $p$-adic Kazhdan-Lusztig hypothesis holds for $\lambda$, then for a Langlands parameter $\varphi$ with infinitesimal parameter $\lambda$, if ${\Pi }_{\varphi }(G)$ contains a generic representation, then $L(s, ϕ, \Ad)$ is regular at $s=1$. 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 ${\varphi }_{\psi }$ is tempered. We also offer some speculation about the relationship between Arthur type representations and singularities in varieties of Langlands parameters defined by Vogan.