Polynomially effective equidistribution for certain unipotent subgroups in quotients of semisimple Lie groups

Abstract: We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of semisimple Lie groups with a polynomial error term. This provides the first infinite family of examples where effective equidistribution, with polynomial error rate, of non-horospherical unipotent subgroups in semisimple quotients is obtained. As applications, we obtain effective estimate on distribution of lattice orbits on homogeneous spaces, as well as an effective version of the Oppenheim conjecture for indefinite quadratic forms with a polynomial error rate in all dimension $d \geq 3$. We also prove a sub-modularity inequality for irreducible representation of connected algebraic group, which is crucial to our proof and is of independent interest.