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

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.