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 GL„(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 GL„(k)
GL„(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 GL„, 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 GL„(k)
contain primitive principal series of GL„(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]).