M. Ratner in a series of papers [Duke Math. J. 63, No. 1, 235–280 (1991; Zbl 0733.22007); Ann. Math. (2) 134, No. 3, 545–607 (1991; Zbl 0763.28012)] proved the Raghunathan conjectures for real Lie groups. Ratner’s theorems describe orbit closures and invariant measures of actions of unipotent subgroups on homogeneous spaces and they have a measure as well as a topological counterpart. M. Ratner [Stud. Math., Tata Inst. Fundam. Res. 14, 167–202 (1998; Zbl 0943.22010)] has conjectured a positive characteristic version of her measure classification theorem.
The paper provides a partial answer to this conjecture by establishing a positive characteristic version of M. Ratner’s classification result for measures invariant under semisimple groups. The approach is based on a servey paper of M. Einsiedler [Jahresber. Dtsch. Math.-Ver. 108, No. 3, 143–164 (2006; Zbl 1136.37021)].