Representations of $p$-adic reductive groups

Abstract

This book presents a part of the (complex) representation theory of $p$-adic reductive groups. Starting from a basis accessible to graduate students, it culminates with the theory of the "Bernstein center" and the Langlands classification of irreducible smooth representations. \vspace{0.5cm} \noindent The book consists of seven chapters, Chapters VI and VII constituting the core of the book. Chapter VI is devoted to the study of the category of smooth representations of a $p$-adic reductive group, establishing among other things the Bernstein decomposition theorem and the description of its center. Chapter VII deals with square-integrable and tempered representations, and the Langlands classification theorem is proved there. \vspace{0.5cm} \noindent The first four chapters are placed in a more general framework and tackle, in order, the study of algebras with idempotents, totally disconnected locally compact spaces and groups, smooth representations of the latter, and particular classes of representations (compact, unitary, square-integrable). Chapter V is a review of the structure results of $p$-adic reductive groups. Appendices give the elements of category theory and some results in algebra used in the text.