Preservation of Unitarizability (Conjecture 3.8)

Show that if π ∈ Irr(G_n) is weakly real and supported in a regular Jantzen decomposition across cuspidal lines X_{ρ_1}, …, X_{ρ_r} and a fixed supercuspidal σ, then π is unitary if and only if each component X_{ρ_i}(π) is unitary for i = 1,…,r.

Background

Jantzen’s decomposition partitions representations along cuspidal lines, allowing reduction of problems to lower rank objects. Tadić conjectured that unitarity is preserved under this decomposition for weakly real representations, which would significantly simplify classification of the unitary dual.

The authors prove an analogue of this for Arthur-type representations (Theorem 6.25) but the original unitarity conjecture remains open.