The Admissible Dual of GL(N) via Compact Open Subgroups. (AM-129), Volume 129 [Paperback]

This work gives a full description of a method for analyzing the admissible complex representations of the general linear groupG=Gl(N,F)of a non-Archimedean local fieldFin terms of the structure of these representations when they are restricted to certain compact open subgroups ofG. The authors define a family of representations of these compact open subgroups, which they callsimple types.The first example of a simple type, the trivial type, is the trivial character of an Iwahori subgroup ofG. The irreducible representations ofGcontaining 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 ofGcontaining a given simple type. This leads to a complete classification of the irreduc-ible smooth representations ofG, 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.