Measure rigidity for solvable group actions in the space of lattices

摘要

We study invariant probability measures on the homogeneous space SLn(R)/SLn(Z) for the action of subgroups of SLn(R) of the form SF where F is generated by one parameter unipotent groups and S is a one parameter R-diagonalizable group normalizing F. Under the assumption that S contains an element with only one eigenvalue less than one counted with multiplicity) and others bigger than one we prove that all the SF invariant and ergodic probability measures on SLn(R)/SLn(Z) are homogeneous.