L-Functions and Representation Theory of p-Adic Groups

Abstract

Abstract The modern theory of automorphic forms as envisioned by Langlands (28, 30] gives an equal weight to representation theory of p-adic reductive groups as that of real groups. While representation theory of real groups is near perfection, a similar statement cannot be made about p-adic groups and in fact there is an abundance of unsolved problems of extreme depth, beauty, and importance that must be answered. The work of Harish-Chandra [11, 45] has already proved that a large part of the theory from real groups carries over, fairly precisely, to the p-adic case. Nothing like this can be said of their discrete series representations and in fact to this date our understanding of them remains poor, except for GLn and certain other classical groups (see the next paragraph).