Strictly small representations and a reduction theorem for the unitary dual (Q2761176)

In previous work [Ann. Math. (2) 148, 1067-1133 (1998; Zbl 0918.22009)] the authors gave a program for classifying the unitary dual for a Lie group \(G\) in Harish-Chandra's class. Under an assumption on the infinitesimal characters of unitary representations, their program reduces the problem of classifying the unitary dual to classifying only the so called unitarily small representations. In that paper, the unitary dual was partitioned into disjoint sets parameterized by a discrete set denoted \(\Lambda_u\). Roughly, to each element \(\lambda_u\in\Lambda_u\) and its centralizer \(G_u\) in \(G\) they attached a set \(\Pi_u^{\lambda_u}(G)\) of unitary irreducible representations of \(G\). In this paper, still assuming the conjecture on infinitesimal characters, the authors refine this partition. In particular, given an irreducible unitary representation \(X\) of \(G\) associated to the subgroup \(G_u\), they find a subgroup \(G_{su}\subseteq G_u\) so that \(X\) is cohomologically induced from some unitary representation of \(G_{su}\). As the class of unitarily small representations is still a large one, this now further reduces the classification problem of the unitary dual of \(G\) to those representations for which \(G_{su}\) is still all of \(G\).