Strictly small representations and a reduction theorem for the unitary dual

Abstract

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