\(p\)-adic norms and quadratic extensions
type: Item
instance of: scholarly article
title: \(p\)-adic norms and quadratic extensions
zbMATH Open document ID: 1206.22009
author: Christophe Cornut
publication date: 2010-02-22T00:00:00Z
full work available at URL: https://eudml.org/doc/10453
review text: Let \(K\) denote a quadratic extension of a \(p\)-adic field \(F\). In the paper under review, the author studies the action of a subgroup \(H\) on the Bruhat-Tits building of a classical \(p\)-adic group \(G\) for the following spherical pairs \((G,H)\): {\parindent=7,5mm \begin{itemize}\item[(i)] \(G = \text{GL}(V)\), \(H=\text{GL}(W)\), where \(W\) denotes a finite dimensional \(K\)-vector space and \(V\) denotes an underlying \(F\)-vector subspace of W. \item[(ii)] \(G= \text{SO}(V, \varphi)\), \(H = U(W, \psi)\), where \((W, \psi)\) is a Hermitian \(K\)-space and \((V, \varphi)\) is a quadratic underlying \(F\)-subspace, with \(\varphi = \text{Tr}_{K / F} \psi\). \item[(iii)] \(G= \text{SO}(V, \varphi)\), \(H = U(W, \psi)\), where \((W, \psi)\) is a Hermitian \(K\)-space, while the underlying quadratic \(F\)-vector space is an \(F\)-hyperplane of a quadratic \(F\)-vector space \((V, \varphi)\). \end{itemize}} The author classifies the orbits of \(H\) and obtains some interesting explicit decompositions of the \(p\)-adic group \(G\), rather similar to a Cartan decomposition, which should have potential applications in harmonic analysis on the homogeneous spherical spaces \(G / H\).
reviewed by: Ivan Matić
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 43A99
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 5673895
zbMATH Keywords: spherical pairs
zbMATH Keywords: \(p\)-adic norms
zbMATH Keywords: quadratic extensions
MaRDI profile type: Publication
OpenAlex ID: W2462973128
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DOI: 10.5802/AIF.2489
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5673895