Sobolev lifting over invariants

32 pages ; We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $\sigma=(\sigma_1,\ldots,\sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map $\bar f : \mathbb R^m \to V$ such that $f = \sigma \circ \bar f$ is of class $C^{d-1,1}$ is locally of Sobolev class $W^{1,p}$ for all $1 \le p < d/(d-1)$. In the case $m=1$ there always exists a continuous choice $\bar f$ for given $f : \mathbb R \to \sigma(V) \subseteq \mathbb C^n$. We give uniform bounds for the $W^{1,p}$-norm of $\bar f$ in terms of the $C^{d-1,1}$-norm of $f$. The result is optimal: in general a lifting $\bar f$ cannot have a higher Sobolev regularity and it even might not have bounded variation if $f$ is in a larger H\"older class.