Planes in four-space and four associated CM points

Abstract

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian $\mathrm{Gr}(2,4)$ and four shapes of lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement, and the second two essentially arise from the accidental local isomorphism between $\mathrm{SO}(4)$ and $\mathrm{SO}(3)\times \mathrm{SO}(3)$. As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings, we prove simultaneous equidistribution of all of these objects under two splitting conditions.