R-groups and Elliptic Representations for Similitude Groups

The tempered spectrum of the similitude groups of non-degenerate symplectic, hermitian, or split orthogonal forms defined over p-adic groups of characteristic zero is studied. The components of representations induced from discrete series of proper parabolic subgroups are classified in terms of R-groups. Multiplicity one is proved. The tempered elliptic spectrum is identified, and the relation between elliptic characters appearing in a given induced representation is determined. Those irreducible tempered representations which are not elliptic and not fully induced from elliptic tempered representations are described. Introduction. We continue our program of studying the explicit description of the tempered spectrum of reductive groups defined over a p-adic field of characteristic zero. The problem can be broken down into two steps. The first step is to classify the discrete series representations of G and of all its Levi subgroups. The second step is to determine the components of those representations which are parabolically induced from a discrete series representation of a proper parabolic subgroup P of G. It is this second step that we shall concern ourselves with. This problem also has two distinct parts. The first is to determine the criteria for an induced representation to be reducible when P is a maximal proper parabolic subgroup. This involves the computation of Plancherel measures. The second part is to use knowledge of the rank one Plancherel measures to construct the Knapp-Stein R-group. This gives one a combinatorial algorithm for determining the structure of the induced representations. Those irreducible tempered representations whose characters fail to vanish on the regular elliptic set are called elliptic tempered representations, and are of particular interest. *Partially supported by NSF Postdoctoral Fellowship DMS9206246 Research at MSRI supported in part by NSF grant DMS902214