A Cartan decomposition for \(p\)-adic loop groups
type: Item
instance of: scholarly article
title: A Cartan decomposition for \(p\)-adic loop groups
zbMATH Open document ID: 0837.22013
DOI: 10.1007/BF01444491
published in: Mathematische Annalen
publication date: 1996-05-13T00:00:00Z
full work available at URL: https://eudml.org/doc/165327
review text: Let \(G\) be a simply connected Chevalley group, \({\mathcal L}_k\) the field of formal Laurent series with coefficients in a \(p\)-adic field \(k\), \({\mathcal O}\) the ring of integers in \(k\), \({\mathcal L}_{{\mathcal O}}\) the ring of elements in \({\mathcal L}_k\) with coefficients in \({\mathcal O}\), \(\overline G= G_{{\mathcal L}_k}\) the group of \({\mathcal L}_k\)-rational points of \(G\), \(\overline K= G_{{\mathcal L}_{\mathcal O}}\) the group of \({\mathcal L}_{{\mathcal O}}\)-rational points of \(G\), and \(\overline A\) the \(k\)-rational points of a maximal split torus in \(G\). The author's purpose is to prove the existence of a Cartan decomposition of \(\overline G\) with respect to \(\overline A\) and \(\overline K\). For technical simplicity the author restricts the considerations to the case when \(G\) corresponds to a simply-laced root system (however, the proof of another result, the existence of maxima, has no such restriction). Actually, the author does not prove the existence of Cartan decompositions for all of \(\overline G\), but only for a subgroup of \(\overline G\) whose elements are bounded.
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 22E65
zbMATH DE Number: 769473
zbMATH Keywords: \(p\)-adic loop groups
zbMATH Keywords: Chevalley group
zbMATH Keywords: Laurent series
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: maximal split torus
zbMATH Keywords: Cartan decomposition
zbMATH Keywords: simply-laced root system
author: H. Garland
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
cites work: The arithmetic theory of loop algebras
cites work: The arithmetic theory of loop groups
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cites work: On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups
cites work: Q3477451
cites work: Integral Bases for Affine Lie Algebras and Their Universal Enveloping Algebras
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description: scientific article; zbMATH DE number 769473