Xu’s closure-ordering conjecture for orbits in local A-packets

Prove that for any connected reductive p-adic group G, any Arthur parameter ψ, and any π ∈ Π_ψ(G), the orbit C_{φ_ψ} associated to the Langlands parameter φ_ψ is contained in the closure of the orbit C_{φ_π}, i.e., C_{φ_ψ} ⊆ \bar{C}_{φ_π}.

Background

This conjecture connects the geometry of Vogan varieties (orbit closures) with the structure of local A-packets, positing a uniform inclusion of orbits tied to parameters in the same packet. It is motivated by ABV-packet analogues and has been verified in specific cases (Sp(2n,F) and split SO(2n+1,F)).

Establishing this inclusion in general would strengthen the interplay between microlocal geometry and Arthur’s classification, offering a geometric constraint on the distribution of representations within A-packets.