Locally unipotent invariant measures and limit distribution of a sequence of polynomial trajectories on homogeneous spaces

Let $G$ be a Lie group and $\mathrm{\Gamma }$ be a lattice in $G$. We introduce the notion of locally unipotent invariant measures on $G\mathrm{/}\mathrm{\Gamma }$. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $G\mathrm{/}\mathrm{\Gamma }$ 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 \cite{shah1994limit}. The proof relies on Ratner’s measure classification theorem, linearization technique for polynomial trajectories near singular sets and a twisting technique of Shah.