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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.

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