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.

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Ian G. Macdonald (11 October 1928 - 8 August 2023) was a British mathematician who made important contributions to algebra, geometry, Lie theory, combinatorics and special functions. He is renowned for the introduction of the Macdonald polynomials and his foundational work on symmetric functions.

Anne-Marie Aubert is a French mathematician, director of Research at CNRS and leader of the team "Automorphic Forms" at the Institut de Mathématiques de Jussieu - Paris Rive Gauche. She is known for her contributions to the representation theory of p-adic groups, character sheaves, Hecke algebras and the Langlands program, in connection with noncommutative geometry.