Strictly small representations and a reduction theorem for the unitary dual

Abstract

To any irreducible unitary representation X X of a real reductive Lie group we associate in a canonical way, a Levi subgroup G s u G_{su} and a representation of this subgroup. Assuming a conjecture of the authors on the infinitesimal character of X X , we show that X X is cohomologically induced from a unitary representation of the subgroup G s u G_{su} . This subgroup is in some cases smaller than the subgroup G u G_{u} that the authors attached to X X in earlier work. In those cases this provides a further reduction to the classification problem.