LOCALLY UNIPOTENT INVARIANT MEASURES AND LIMIT DISTRIBUTION OF A SEQUENCE OF POLYNOMIAL TRAJECTORIES ON HOMOGENEOUS SPACES

Let G be a Lie group, and let Gamma be a lattice in G. We introduce the notion of locally unipotent invariant measures on G/Gamma. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on G/Gamma is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in [15].The proof relies on Ratner's measure classification theorem, a lineariza-tion technique for polynomial trajectories near singular sets, and a twisting technique of Shah.