REDUCIBILITY FOR SU n AND GENERIC ELLIPTIC REPRESENTATIONS

Introduction. The problem of classifying the tempered spectrum of a connected reductive quasi-split group, defined over a local field, F, of characteristic zero, consists of three parts. The first is to classify the discrete series representations of any Levi subgroup. The second step is to understand the rank one Plancherel measures, which is equivalent to understanding the reducibility of those representations parabolically induced from a discrete series of maximal Levi subgroups. The third step is to understand the structure of representations parabolically induced from discrete series representations of an arbitrary parabolic subgroup, using the second step and the combinatorial theory of the Knapp-Stein R–group. We address this third step here for the case where the group in question is the quasi-split special unitary group.