  We introduce the notion of the weak tracial approximate representability of a
discrete group action on a unital $C^*$-algebra which could have no projections
like the Jiang-Su algebra $\mathcal{Z}$. Then we show a duality between the
weak tracial Rokhlin property and the weak tracial approximate
representability. More precisely, when $G$ is a finite abelian group and
$\alpha:G\curvearrowright A$ is a group action on a unital simple infinite
dimensional $C^*$-algebra, we prove that
  1. $\alpha$ has the weak tracial Rokhlin property if and only if
$\hat{\alpha}$ has the weak tracial approximate representability.
  2. $\alpha$ has the weak tracial approximate representability if and only if
$\hat{\alpha}$ has the weak tracial Rokhlin property.
