Representations of groups over close local fields
type: Item
instance of: scholarly article
title: Representations of groups over close local fields
zbMATH Open document ID: 0634.22010
DOI: 10.1007/BF02792537
author: David Kazhdan
published in: Journal d'Analyse Mathématique
publication date: 1986-01-01T00:00:00Z
review text: Let F and F' be nonarchimedean local fields, with rings of integers O and O', respectively, and \(P\subset O\) and P'\(\subset O'\) the maximal ideals. Say that F and F' are m-close if the quotient rings \(O/P^ m\) and \(O'/P^{'m}\) are isomorphic. Let \({\mathbb{G}}\) be a reductive Z-group, assumed split for convenience. For any ring A, set \(G_ A={\mathbb{G}}(A)\). For any positive integer l, let \(A_ l\) be the quotient \(O/P^ l\), and let \(K_ l(F)\) be the congruence subgroup of \(G_ O mod P^ l\). Identify \(G_{A_ l}\) with \(G_ O/K_ l(F)\). Let \(H_ l(G,K)\) be the Hecke algebra of \(K_ l(F)\)-biivariant measures, under convolution, on G. The main theorem is that given a local nonarchimedean field F and a positive integer l, there exists an \(m\geq l\) such that, if F' is m-close to F, then the natural morphism \(H_ l(G,F)\to H_ l(G,F')\) is an algebra isomorphism. As a corollary, if F is a local field of positive characteristic, and l is a positive integer, then there exists a local field F' of characteristic zero such that the Hecke algebras \(H_ l(G,F)\) and \(H_ l(G,F')\) are isomorphic. Since irreducible \(H_ l\)- modules are in one-to-one correspondence with irreducible representations of G which have non-zero \(K_ l\)-fixed vectors, this result can be used, for example, to extend results proved for groups over fields of characteristic zero, to groups over fields of positive characteristic. The present paper contains the following application. In the article reviewed above, the author shows for groups over fields of characteristic zero, that the characters of tempered admissible representations of G are dense in the space of all invariant distributions on G. The corresponding result for groups over fields of positive characteristic is shown using a result of Bernstein: The following are equivalent. (1) Suppose h is an element of the Hecke algebra H(G) for which \(<\pi,h>=0\) for all irreducible representations \(\pi\) of G. Then h is annihilated by all …
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 22E20
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 4031985
zbMATH Keywords: nonarchimedean local fields
zbMATH Keywords: reductive Z-group
zbMATH Keywords: congruence subgroup
zbMATH Keywords: biivariant measures
zbMATH Keywords: Hecke algebras
zbMATH Keywords: irreducible representations
zbMATH Keywords: characters of tempered admissible representations
zbMATH Keywords: invariant distributions
MaRDI profile type: Publication
cites work: Hecke rings of congruence subgroups
cites work: Cuspidal geometry of p-adic groups
description: scientific article; zbMATH DE number 4031985