Polynomially Effective Equidistribution for Some Higher Dimensional Unipotent Subgroups

Let G be a perfect Lie group, Γ be a lattice in G and U be a unipotent subgroup of G. A celebrated theorem of Ratner says that for any x in G/Γ the orbit U.x is equidistributed in a periodic orbit of some subgroup U≤L≤G. Establishing a quantitative version of Ratner's theorem has been long sought after. If U is a horospherical subgroup of G, the question is well-studied. If U is not a horospherical subgroup, this question is far less understood. Recently, Lindenstrauss, Mohammadi, Wang and Yang established fully quantitative and effective equidistribution results for orbits of one-parameter (non-horospherical) unipotent groups in a wide variety of cases. In this talk, we will discuss a recent equidistribution theorem for some higher dimensional unipotent subgroups. Our results in particular provide effective equidistribution theorems for orbits of maximal unipotent subgroups of SO(p,q) in SLn(ℝ)/SLn(ℤ) for all n=p+q. If time permits, we will also discuss a submodularity inequality in irreducible representations, which is a key ingredient of the proof and is of independent interest.