A class of multipliers for $$\mathcal{W}^ \bot $$.

Let $$\mathcal{W}^ \bot $$ denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let $$\mathcal{M}(\mathcal{W}^ \bot )$$ be the class of multipliers for $$\mathcal{W}^ \bot $$ , i.e. ergodic transformations whose all ergodic joinings with any element of $$\mathcal{W}^ \bot $$ are also in $$\mathcal{W}^ \bot $$ . Fix an ergodic rotation T, a mildly mixing action S of a locally compact second countable group G and an ergodic cocycle ϕ for T with values in G. The main result of the paper is a sufficient (and also necessary by [LeP] when G is countable Abelian and S is Bernoullian) condition for the skew product build from T, ϕ and S to be an element of $$\mathcal{M}(\mathcal{W}^ \bot )$$ . Moreover, the self-joinings of such extensions of T are described with an application to study semisimple extensions of rotations. [ABSTRACT FROM AUTHOR]