The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

This work gives a full description of a method for analyzing the admissible complex representations of the general linear group "G" = "Gl(N, F)" of a non-Archimedean local field "F" in terms of the structure of these representations when they are restricted to certain compact open subgroups of "G." The authors define a family of representations of these compact open subgroups, which they call "simple types." The first example of a simple type, the "trivial type," is the trivial character of an Iwahori subgroup of "G." The irreducible representations of "G" containing the trivial simple type are classified by the simple modules over a classical affine Hecke algebra. Via an isomorphism of Hecke algebras, this classification is transferred to the irreducible representations of "G" containing a given simple type. This leads to a complete classification of the irreduc-ible smooth representations of "G," including an explicit description of the supercuspidal representations as induced representations. A special feature of this work is its virtually complete reliance on algebraic methods of a ring-theoretic kind. A full and accessible account of these methods is given here.