Limits of discrete series with infinitesimal character zero

Abstract:

For a connected linear semisimple Lie group $G$, this paper considers those nonzero limits of discrete series representations having infinitesimal character 0, calling them totally degenerate. Such representations exist if and only if $G$ has a compact Cartan subgroup, is quasisplit, and is acceptable in the sense of Harish-Chandra. Totally degenerate limits of discrete series are natural objects of study in the theory of automorphic forms: in fact, those automorphic representations of adelic groups that have totally degenerate limits of discrete series as archimedean components correspond conjecturally to complex continuous representations of Galois groups of number fields. The automorphic representations in question have important arithmetic significance, but very little has been proved up to now toward establishing this part of the Langlands conjectures. There is some hope of making progress in this area, and for that one needs to know in detail the representations of $G$ under consideration. The aim of this paper is to determine the classification parameters of all totally degenerate limits of discrete series in the Knapp–Zuckerman classification of irreducible tempered representations, i.e., to express these representations as induced representations with nondegenerate data. The paper uses a general argument, based on the finite abelian reducibility group $R$ attached to a specific unitary principal series representation of $G$. First an easy result gives the aggregate of the classification parameters. Then a harder result uses the easy result to match the classification parameters with the representations of $G$ under consideration in representation-by-representation fashion. The paper includes tables of the classification parameters for all such groups $G$.