The Admissible Dual of GL _N Via Restriction to Compact Opent Subgroups

Abstract

Let G be a reductive group over a p-adic field F. Then, as with reductive groups over any field, it is natural to cast the representation theory of G in terms of parabolic induction. This leads to the notion of supercuspidal representation and, in the case of GLn, to the classification of irreducible (admissible) representations given in the work of Bernstein-Zelevinski [BZ], [Z]. On the other hand, the fact that G is a totally disconnected, locally compact group accounts for the existence of open, compact modulo center subgroups of G which in turn has a strong influence on its representation theory. In particular, one is led to consider the possibility that supercuspidal representations may be constructed by induction from such subgroups (see [Ku2] for historical background) and, more generally, to inquire into the possibility of classifying admissible representations of G by considering the subrepresentations they may have when restricted to such subgroups (the possibility of classifying the admissible dual in this fashion was first raised in [H].) In what follows, we report on recent progress in this direction in the case G = GLn(F)\ we begin with some general background.

The first author was supported in part by SERC grant GR/E 47650. The second author was supported in part by NSF Grant DMS-8704194 and by SERC grant GR/F 73366. Borth author wish to thank the Institute for Advanced Study for their hospitality during Academic Year 1988-1989. This visit was supportes in part by NSF Grant DMS 8610730