Arthur representations and unitary dual for classical groups (2410.11806v2)

Abstract: In this paper, we propose a new conjecture (Conjecture 1.1) on the structure of the unitary dual by means of the Arthur representations for general reductive algebraic groups defined over any non-Archimedean local field of characteristic zero. We also propose a conjecture (Conjecture 1.2) refining Conjecture 1.1 for representations of good parity. The relations among the two conjectures and special families of representations are explained in Figure 1. The main results include a partial approval of Conjecture 1.1 and the verification of Conjectures 1.1 and 1.2 for representations of corank at most 3 for symplectic or split odd special orthogonal groups, based on Tadi{\'c}'s classification (Theorem 1.4). To prove the main results, we develop new algorithms to determine whether a given irreducible representation is of Arthur type and give an inductive approach to classify the family of unitary representations that are of Arthur type for classical groups. We explicate this approach towards the unitary dual problem for representations of corank 3 and several new families of representations.