An Assortment of Results on Representations of GL₂(k)

Abstract

Let k be a p-adic field, o its integers. My aim in these notes has been to collect in one convenient place most of the results I need in [5] concerning the representations of GL₂(k). My choice of which results to include was determined by this, but my choice of which proofs of these results to follow was determined by two aims: to use methods as generally applicable as possible to other reductive p-adic groups besides GL₂(k) GL₂(k), so as to convey some of the flavor of the subject to an audience who for the most part knew very little about it, and to relate to the classical theory of modular forms as closely as possible. Thus, I have gone into a great deal of detail about Hecke algebras. I have avoided using the Kirillov-Whittaker model of representations of GL₂, partly because Deligne has already covered it, partly because Jacquet-Langlands cannot be drastically improved on, but mostly because the proofs of the results which I need - for example, that only principal series of GL₂(k) contain primitive principal series of GL₂(o) - seem to me less clear if one uses it (although this particular result has, in fact, a proof using the Kirillov-Whittaker model. See [4]).

The author’s travel expences for this conference were paid for by a grant from the National Research Council of Canada.