Equidistribution of rational subspaces and their shapes

Abstract To any k -dimensional subspace of $\mathbb {Q}^n$ one can naturally associate a point in the Grassmannian $\mathrm {Gr}_{n,k}(\mathbb {R})$ and two shapes of lattices of rank k and $n-k$ , respectively. These lattices originate by intersecting the k -dimensional subspace and its orthogonal with the lattice $\mathbb {Z}^n$ . Using unipotent dynamics, we prove simultaneous equidistribution of all of these objects under congruence conditions when $(k,n) \neq (2,4)$ .