The closure ordering conjecture on local Arthur packets of classical groups

Abstract We prove the closure ordering conjecture on the local 𝐿-parameters of representations in local Arthur packets of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">G</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>Sp</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>SO</m:mi> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mrow> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Π</m:mi> <m:mi>ψ</m:mi> </m:msub> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">G</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> \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.