Spherical Functions on a Group of p-adic Type by Macdonald, Ian G.; Aubert, Anne-Marie;

This is a new, updated edition of a foundational text on the representation theory of p-adic groups.

The book develops the theory of spherical functions for reductive groups defined over nonarchimedean local fields. It provides explicit formulas, studies their properties (positivity, normalization, etc.), and describes a pioneering construction of the spherical transform and the Plancherel formula. This theory underlies the modern theory of affine Hecke algebras, unramified representations of p-adic groups, and the local Langlands program.

This augmented and annotated edition makes a standard reference widely available to contemporary researchers in the representation theory of p-adic groups, automorphic forms, and harmonic analysis on locally compact groups.