  An isotopic to the identity map of the $2$-torus, that has zero rotation
vector with respect to an invariant ergodic probability measure, has a fixed
point by a theorem of Franks. We give a version of this result for nilpotent
subgroups of isotopic to the identity diffeomorphisms of the $2$-torus. In such
a context we guarantee the existence of global fixed points for nilpotent
groups of irrotational diffeomorphisms. In particular we show that the derived
group of a nilpotent group of isotopic to the identity diffeomorphisms of the
$2$-torus has a global fixed point.
