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In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic $p\ne <\#comment/>2$. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply a new idea by considering a certain sum of partial Bessel functions to overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups. 

Abstract We prove the closure ordering conjecture on the local 𝐿-parameters of representations in local Arthur packets of ${\mathrm{G}}_n={\mathrm{Sp}}_{2n},{\mathrm{SO}}_{2n+1}$\mathrm{G}_{n}=\mathrm{Sp}_{2n},\mathrm{SO}_{2n+1}over a non-Archimedean local field of characteristic zero.Precisely, given any representation 𝜋 in a local Arthur packet ${\mathrm{\Pi }}_{\psi }$\Pi_{\psi}, the closure of the local 𝐿-parameter of 𝜋 in the Vogan variety must contain the local 𝐿-parameter corresponding to 𝜓.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 ${\mathrm{G}}_n$\mathrm{G}_{n}, hence give a new proof of the enhanced Shahidi conjecture.Finally, 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 is of independent interest.