\(l\)-modular representations of a \(p\)-adic reductive group with \(l\neq p\)
type: Item
instance of: scholarly article
title: \(l\)-modular representations of a \(p\)-adic reductive group with \(l\neq p\)
zbMATH Open document ID: 0859.22001
author: Marie-France Vigneras
published in: Progress in Mathematics
publication date: 1996-08-29T00:00:00Z
review text: Let \(F\) be a \(p\)-adic field and \(G\) the group of \(F\)-rational points of a reductive connected group over \(F\). Then \(G\) is a totally disconnected locally compact, or, what amounts to the same, a locally profinite group. For a complex representation \(V\) one usually considers the ``skeleton'' of smooth vectors \(V^\infty\), consisting of all vectors with open stabilizer in \(G\). The algebraic representation \(V^\infty\) then essentially determines \(V\) and it is well known that in this way the representation theory of \(G\) can be made an essentially algebraic theory. Since the topology of the complex numbers does not show up anymore in the notion of smooth representations, the latter can be considered over any ground field. By obvious reasons it seems natural to restrict to an algebraically closed ground field so that only the characteristic really plays a role. The present monograph features an introduction to the theory of \(\overline{\mathbb{F}}_l\)-valued smooth representations, where \(l\neq p\) is a prime number, \(\mathbb{F}_l\) is the prime field with \(l\) elements and \(\overline{F}_l\) is an algebraic closure of \(\mathbb{F}_l\). As a by-product purely algebraic proofs are given for a number of results in the theory of (complex) smooth representations which originally have been proven by means of analytic methods. Chapter I gives the general theory of smooth representations of an arbitrary locally profinite group over any commutative ring. The main topics here are induction of representations and the general theory of Hecke algebras. In chapter II the theory is specialized to reductive \(p\)-adic groups. One encounters parabolic induction, cuspidal and supercuspidal representations and the theory of minimal types for \(\overline{\mathbb{F}}_l\)-valued representations. This theory requires some knowledge about the Bruhat-Tits building, the latter only being cited in the present book. In the last chapter we are given the theory of types by How…
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 921901
zbMATH Keywords: locally profinite group
zbMATH Keywords: complex representation
zbMATH Keywords: smooth representations
zbMATH Keywords: Hecke algebras
zbMATH Keywords: reductive \(p\)-adic groups
zbMATH Keywords: cuspidal and supercuspidal representations
zbMATH Keywords: Bruhat-Tits building
zbMATH Keywords: theory of types
reviewed by: Anton Deitmar
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 921901