Finitely differentiable invariants
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.
finitely differentiable \(\Gamma\)-invariant functions
scientific article; zbMATH DE number 1336826