Finitely differentiable invariants
type: Item
instance of: scholarly article
title: Finitely differentiable invariants
zbMATH Open document ID: 0930.58004
DOI: 10.1007/PL00004677
author: Matthias Rumberger
published in: Mathematische Zeitschrift
publication date: 2000-02-09T00:00:00Z
review text: The algebra of polynomials on \(\mathbb{R}^m\) which are invariant with respect to a reductive real algebraic group \(\Gamma\) is finitely generated, say by \(\rho_1,\dots, \rho_k\). A classical result by \textit{G. W. Schwarz} [Topology 14, 63-68 (1975; Zbl 0297.57015)] states that, if \(\Gamma\) is compact, each smooth (i.e. \(C^\infty\)) \(\Gamma\)-invariant function can be represented by a function on \(\mathbb{R}^k\) which is also smooth. Thus, there is a map \({\mathcal J}: C^\infty(\mathbb{R}^m)^\Gamma\to C^\infty(\mathbb{R}^k)\) such that \({\mathcal J}(f)(\rho_1(x),\dots, \rho_k(x))= f(x)\) for all \(f\in C^\infty(\mathbb{R}^m)^\Gamma\). Indeed, \({\mathcal J}\) can be assumed to be continuous [\textit{J. W. Mather}, Topology 16, 145-155 (1977; Zbl 0376.58002)]. In the paper, finitely differentiable \(\Gamma\)-invariant functions are considered. In this case a loss of differentiability may occur when factoring out the symmetry. The main result states that a number \(q\) exists such that a map \({\mathcal J}: C^{qn}(\mathbb{R}^m)^\Gamma\to C^n(\mathbb{R}^k)\) can be found with the property stated above. Moreover, as in the smooth case, this theorem extends to reductive groups. The result applies to equivariant maps and vector fields.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 1336826
zbMATH Keywords: finitely differentiable \(\Gamma\)-invariant functions
zbMATH Keywords: equivariant maps
zbMATH Keywords: equivariant vector fields
MaRDI profile type: Publication
full work available at URL: pl00004677
OpenAlex ID: W2071107554
description: scientific article; zbMATH DE number 1336826