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Let X be a smooth algebraic variety endowed with an action of a finite group G such that there exists a geometric quotient Π X : X → X/G. We characterize rational tensor fields τ on X/G such that the pull back of τ is regular on X: these are precisely all τ such that \(di{v_{{R_{X/G}}}}(\tau ) \geqslant 0\) where RX/G is the reflection divisor of X/G and \(di{v_{{R_{X/G}}}}(\tau )\) is the RX/G-divisor of τ. We give some applications, in particular to a generalization of Solomon’s theorem. In the last section we show that if V is a finite dimensional vector space and G a finite subgroup of GL(V), then each automorphism Ψ of V/G admits a biregular lift φ : V → V provided that Ψ maps the regular stratum to itself and Ψ*(RX/G) = RX/G.

Invariant Tensor Fields and Orbit Varieties for Finite Algebraic Transformation Groups