Unitary Dual Conjecture for Classical Groups (Conjecture 1.1)

Establish the equality Π_A(G) = Π_{A+,u}(G) = Π_u(G) for classical split symplectic or split odd special orthogonal groups G over non-Archimedean local fields of characteristic zero, where Π_A(G) denotes the closure of Arthur-type packets as defined via local Arthur parameters and Π_{A+,u}(G) denotes the closure under complementary series and unitary parabolic induction, and Π_u(G) is the full unitary dual.

Background

The paper studies the relationship between Arthur representations ΠA(G), the extended Arthur-type set Π{A+,u}(G), and the full unitary dual Π_u(G) for split classical groups G over non-Archimedean local fields of characteristic zero. Arthur packets are expected to be subsets of the unitary dual, but examples show subtleties that necessitate defining closures under specific unitary constructions.

Conjecture 1.1 proposes that these closures suffice to recover the entire unitary dual for classical groups, providing a representation-theoretic characterization of unitary duals via Arthur theory. The authors verify the conjecture for several families and for corank ≤ 3, but the general case remains open.