  Translators in the special linear group $SL(2,\mathbb{R})$ are surfaces whose
mean curvature $H$ and unit normal vector $N$ satisfy $H=\langle N,X\rangle$,
where $X$ is a fixed Killing vector field. In this paper we study and classify
those translators that are invariant by a one-parameter group of isometries. By
the Iwasawa decomposition, there are three types of such groups. The dimension
of the Killing vector fields is $4$ and an exhaustive discussion is done for
each one of the Killing vector fields and each of the invariant surfaces. In
some cases, explicit parametrizations of translators are obtained.
