Tadić’s Conjecture on Critical-Type Unitarity (Conjecture 1.5)

Establish that, for any classical group G over non-Archimedean local fields of characteristic zero, Π_{A,crit}(G) = Π_{u,crit}(G) and Π_iso(G) ⊆ Π_{u,crit}(G), where Π_{A,crit}(G) and Π_{u,crit}(G) are the Arthur-type and unitary subsets of critical-type representations and Π_iso(G) denotes the isolated unitary representations.

Background

Critical-type representations are a special subfamily of good-parity representations designed to capture delicate boundary behaviors (e.g., complementary series endpoints). Tadić conjectured that critical-type Arthur representations coincide with critical-type unitary ones and that isolated unitary representations sit inside the critical-type unitary set.

The authors relate this conjecture to their broader Conjecture 1.2, noting that equality of the critical-type sets would follow from Conjecture 1.2 and discussing evidence in several cases.