The absolute continuous spectrum of skew products of compact Lie groups

Let $X$ and $G$ be compact Lie groups, $F_1:X\to X$ the time-one map of a $C^{\mathrm{\infty }}$ measure-preserving flow, $\varphi :X\to G$ a continuous function and $\pi$ a finite-dimensional irreducible unitary representation of $G$. Then, we prove that the skew products $$T_{\varphi }:X\times G\to X\times G,(x,g)\mapsto (F_1(x),g\varphi (x)),$$ have purely absolutely continuous spectrum in the subspace associated to $\pi$ if $\pi \circ \varphi$ has a Dini-continuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of $\pi \circ \varphi$ has nonzero determinant. This result provides a simple, but general, criterion for the presence of an absolutely continuous component in the spectrum of skew products of compact Lie groups. As an illustration, we consider the cases where $F_1$ is an ergodic translation on ${\text{T}}^d$ and $X\times G={\text{T}}^d\times {\text{T}}^{d^{\mathrm{\prime }}}$, $X\times G=\T^d\times\SU(2)$and$X\times G=\T^d\times\U(2)$. Our proofs rely on recent results on positive commutator methods for unitary operators.