Good-Parity Arthur-Type Equivalence (Conjecture 1.2)

Prove that, for any classical split symplectic or split odd special orthogonal group G over a non-Archimedean local field of characteristic zero, a representation of good parity is of Arthur type if and only if it is unitary; equivalently, show Π_{A,gp}(G) = Π_{A+,gp}(G) = Π_{u,gp}(G), where Π_{A,gp}(G) = Π_gp(G) ∩ Π_A(G), Π_{A+,gp}(G) = Π_gp(G) ∩ Π_{A+,u}(G), and Π_{u,gp}(G) = Π_gp(G) ∩ Π_u(G).

Background

The authors introduce the notion of good parity and define the subsets Π{A,gp}(G), Π{A+,gp}(G), and Π_{u,gp}(G) that restrict Arthur-type and unitary representations to good parity. Conjecture 1.2 refines Conjecture 1.1 by asserting an exact match among these sets for good-parity representations.

They provide algorithms and partial results validating the conjecture for several classes, including representations of corank ≤ 3, generic, and unramified families, but the general assertion remains unproven.

References

Conjecture 1.2. For a classical group G, the following holds:

(1.1) Π (G) = Π (G) = Π (G). A,gp A,gp u,gp

In other words, for any π ∈ Π (gp, π is of Arthur type if and only if it is unitary.

— Arthur representations and unitary dual for classical groups (2410.11806 - Hazeltine et al., 15 Oct 2024) in Conjecture 1.2, Section 1 (Introduction)