Let Xk,d denote the space of rank-k lattices in Rd. Topological and statistical properties of the dynamics of discrete subgroups of G=SL(d,R) on Xd,d were described in the seminal works of Benoist-Quint. A key step/result in this study is the classification of stationary measures on Xd,d. Later, Sargent-Shapira initiated the study of dynamics on the spaces Xk,d. When k ≠ d, the space Xk,d is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure μ which is Zariski-dense in a copy of SL(2,R) in G, we give a classification of μ-stationary measures on Xk,d and prove corresponding equidistribution results. In contrast to the results of Benoist-Quint, the type of stationary measures that μ admits depends strongly on the position of SL(2,R) relative to parabolic subgroups of G. I will start by reviewing preceding major works and ideas. The talk will be accessible to a broad audience. Joint work with Alexander Gorodnik and Jialun Li.