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Finitely differentiable invariants. (English) Zbl 0930.58004
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 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 [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.

58C25 Differentiable maps on manifolds
57S20 Noncompact Lie groups of transformations
57S15 Compact Lie groups of differentiable transformations
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