This book presents a part of the theory of (complex) representations of $p$-adic reductive groups. Starting from fundamentals accessible to graduate students, it culminates with the "Bernstein center" theory and the Langlands classification of smooth irreducible representations.

This book contains seven chapters. Chapters VI and VII are at the heart of the book. Chapter VI deals with the study of the category of smooth representations of a $p$-adic reductive group and the author establishes, among other things, Bernstein's decomposition theorem and the description of the center. Chapter VII deals with square integrable and temperate representations and contains the proof of Langland's classification theorem.

The first four chapters are more general and deal with: the study of algebras of idempotents, the one of locally compact totally discontinuous spaces and groups, smooth representations of the latter and specific representation classes (compact, unitary, square integrable). Chapter V is a reminder of structural results for reductive $p$-adic groups.

An appendix provides category theory notions necessary for reading this text.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in representations of $p$-adic reductive groups.

Table of Contents

• Algèbres à idempotents
• Espaces et groupes totalement discontinous
• Représentations des groupes totalement discontinous
• Représentations compactes, de carré intégrable, unitaires
• Structure des groupes réductifs $p$-adiques
• Représentations des groupes réductifs $p$-adiques
• Classification de Langlands
• Éléments de théorie des catégories
• Théorème d'Amitsur et corollaires
• Algèbre linéaire
• Bibliographie