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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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