On describing invariant subspaces of the space of smooth functions on a homogeneous manifold
type: Item
instance of: scholarly article
title: On describing invariant subspaces of the space of smooth functions on a homogeneous manifold
zbMATH Open document ID: 1050.43008
DOI: 10.1023/B:ACAP.0000024205.20730.63
author: Sergei S. Platonov
published in: Acta Applicandae Mathematicae
publication date: 2004-08-20T00:00:00Z
review text: Let \(G\) be a Lie group acting on a smooth manifold \(M.\) For every \(g\in G\) and any complex-valued function \(f\) on \(M,\) define \((\pi(g)f)(x)=f(g^{-1}x).\) A locally convex topological vector space \(\mathcal{F}\) of complex-valued functions (or distributions) defined on \(M\) is said to be \(\pi\)-invariant if for any \(f\in\mathcal{F}\) and any \(g\in G,\) \(\pi(g)f\in\mathcal{F}\) and the map \(g\mapsto\pi(g)f\) is continuous from \(G\) to \(\mathcal{F}.\) A vector subspace \(H\subset\mathcal{F}\) is called an invariant subspace if it is closed and \(\pi\)-invariant. The author gives a survey of some results concerning the description of invariant subspaces of function spaces on homogeneous manifolds.
reviewed by: Roman Urban
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 2093667
zbMATH Keywords: invariant subspaces
zbMATH Keywords: symmetric spaces
zbMATH Keywords: representations of Lie groups
zbMATH Keywords: quasi-regular representation
MaRDI profile type: Publication
full work available at URL: b:acap.0000024205.20730.63
OpenAlex ID: W2013961053
description: scientific article; zbMATH DE number 2093667