Equidistribution in measure-preserving actions of semisimple groups : case of $S{L}_2(\mathbb{R})$

We prove pointwise convergence for the semi-radial averages on $G=S{L}_2(\mathbb{R})$ given by ${\int }_t^{t+1}{m}_K\ast {\delta }_{a_s}ds$ (and similar variants), acting on $K$-finite $L^p$-functions in a probability-measure-preserving action of the group, for $p>1$.