We consider skew products transformations

where X is a compact manifold with probability measure  G is a compact Lie group with Lie algebra   is the time-one map of a measure-preserving flow, and  is a cocycle. Then we define the degree of φ as a suitable function  and show that the degree of φ transforms in a natural way under Lie group homomorphisms and under the relation of -cohomology, and we explain how it generalises previous definitions. For each finite-dimensional irreducible unitary representation π of G, and  the Lie algebra of  we define in an analogous way the degree of  as a suitable function  If  is uniquely ergodic and the functions  are diagonal, or if  is uniquely ergodic, then the degree of π reduces to a constant in  given by an integral (average) over X. As a by-product, we obtain that there is no uniquely ergodic skew products  with nonzero degree if G is a connected semisimple compact Lie group.

Next, we show that  is mixing in the orthocomplement of the kernel of and under some additional assumptions, we show that  has purely absolutely continuous spectrum in that orthocomplement if  is strictly positive. Summing up these individual results for each representation π, one obtains a global result for the mixing property and the absolutely continuous spectrum of  As an application, we present in more detail four explicit cases: when G is a torus,   and 

Our proofs rely on new results on positive commutator methods for unitary operators in Hilbert spaces.

cocycles, skew products, Lie groups, degree, mixing, continuous spectrum, commutator methods.