DSNT Seminar: Stationary measures for $\SL_2(\R)$-actions on homogeneous bundles over flag varieties

Abstract: Let $X_{k,d}$ denote the space of rank-$k$ lattices in $\R^d$. Topological and statistical properties of the dynamics of discrete subgroups of $G=\SL_d(\R)$ on $X_{d,d}$ were described in the seminal works of Benoist--Quint. A key step/result in this study is the classification of stationary measures on $X_{d,d}$. Later, Sargent--Shapira initiated the study of dynamics on the spaces $X_{k,d}$. When $k \neq d$, the space $X_{k,d}$ is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure $\mu$ Zariski-dense in a copy of $\SL_2(\R)$ in $G$, we give a classification of $\mu$-stationary measures on $X_{k,d}$ and prove corresponding equidistribution results. In contrast to the results of Benoist--Quint, the type of stationary measures that $\mu$ 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. Joint work with Alexander Gorodnik and Jialun Li.
 

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