  A root of unity limit of the $q$-deformed Virasoro algebra is considered. The
$\widehat{sl}(2)_k$ current algebra and the integral formulas of the solutions
of the KZ equations can be realized by the $q$-deformed boson at the limit and
an additional boson. We explicitly construct the integral representation of the
four-point blocks with a $\mathcal{K}$-operator insertion.
