The existence of $S^1\times C_p$-maps between representation spheres and its applications

We show the existence of $S^1\times C_p$-maps between certain representation spheres. As an application, we show that, in the family of abelian compact Lie groups, a group $G$ has the weak Borsuk-Ulam property (in the sense of Bartsch) if and only if $G$ is either a finite abelian $p$-group or a $k$-torus.