The closure ordering conjecture on local Arthur packets of classical groups

In this paper, we prove the closure ordering conjecture on the local $L$-parameters of representations in local Arthur packets of $\mathrm{G}_n=\mathrm{Sp}_{2n}, \mathrm{SO}_{2n+1}$ over a non-Archimedean local field of characteristic zero. Precisely, given any representation $\pi$ in a local Arthur packet $\Pi_{\psi}$, the closure of the local $L$-parameter of $\pi$ in the Vogan variety must contain the local $L$-parameter corresponding to $\psi$. This conjecture reveals a geometric nature of local Arthur packets and is inspired by the work of Adams, Barbasch, and Vogan, and the work of Cunningham, Fiori, Moussaoui, Mracek, and Xu, on ABV-packets. As an application, for general quasi-split connected reductive groups, we show that the closure ordering conjecture implies the enhanced Shahidi conjecture, under certain reasonable assumptions. This provides a framework towards the enhanced Shahidi conjecture in general. We verify these assumptions for $\mathrm{G}_n$, hence give a new proof of the enhanced Shahidi conjecture. At last, we show that local Arthur packets cannot be fully contained in other ones, which is in contrast to the situation over Archimedean local fields and has its own interests.