Topological central extensions of semi-simple groups over local fields. I, II
type: Item
instance of: scholarly article
title: Topological central extensions of semi-simple groups over local fields. I, II
zbMATH Open document ID: 0552.20025
DOI: 10.2307/2006967
author: Gopal Prasad
author: M. S. Raghunathan
published in: Annals of Mathematics. Second Series
publication date: 1984-01-01T00:00:00Z
review text: A central extension \(U\) of a group \(G\) is called universal if any central extension \(E\) of \(G\) factors uniquely through a homomorphism \(U\to E\) over \(G\); such an extension exists if and only if \(G\) is equal to its commutator subgroup \(G'=(G,G)\) and then it is unique up to a \(G\)-isomorphism. In this case, such a universal central extension is described from a free group \(F\) with image \(G\) as the commutator group of \(F/(K,F)\) with \(K\) the kernel of \(F\to G\), and the kernel of the projection of this extension onto \(G\) is naturally the second homology group \(H_2(G,{\mathbb{Z}})\). When \(G\) is a topological group, one has to consider the topological central extensions; when the group is second countable, the classes of central extensions of \(G\) by a commutative group are classified by the group \(H^2(G,C)\) of measurable classes of 2-cocycles on \(G\) with values in \(C\). For the group \(G=SL_ 2(F)\) over a field \(F\), the restriction to \(F^{\times}\) embedded in \(G\) by the co-root \(t\mapsto\left(\begin{smallmatrix} t&0\\ 0&t^{-1}\end{smallmatrix}\right)\) of a 2-cocycle on \(G\) gives a 2-cocycle on \(F^{\times}\), and using the presentation given by R. Steinberg, C. Moore and H. Matsumoto described all the co-cycles obtained, using a short co-root and also those coming from the case when \(F\) is a local field and the extensions are topological. For a non-archimedean local field, this gives \(H_ 2(G,{\mathbb{Z}})\) as the group \(\mu(F)\) of roots of unity in \(F\), and \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) as its dual \(\mu(F)^{\wedge}\) sent in \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) through the norm residue symbol and the above restriction from \(G\) to \(F^{\times}\). H. Matsumoto proved that the same result holds when \(G\) is the group of \(F\)-points, \(F\) a non-archimedean local field, of a connected, simply connected, simple, split algebraic group over \(F\), and the case of quasi-split groups was done by V. V. Deodhar a…
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 22E20
Mathematics Subject Classification ID: 14E20
zbMATH DE Number: 3879147
zbMATH Keywords: universal central extensions
zbMATH Keywords: topological central extensions
zbMATH Keywords: local field
zbMATH Keywords: roots of unity
zbMATH Keywords: norm residue symbol
zbMATH Keywords: quasi-split groups
zbMATH Keywords: rational points
zbMATH Keywords: non-archimedean local fields
zbMATH Keywords: connected, simply-connected absolutely simple isotropic algebraic groups
zbMATH Keywords: Bruhat-Tits buildings
zbMATH Keywords: cohomology
zbMATH Keywords: parahoric subgroups
zbMATH Keywords: Kneser-Tits conjecture
zbMATH Keywords: unipotent radicals
zbMATH Keywords: parabolic subgroups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2006967
OpenAlex ID: W4235907238
description: scientific article; zbMATH DE number 3879147