Equidistribution of rational subspaces and their shapes (Q6624536)

The restriction of a positive-definite integral quadratic form \(Q\) on \(\mathbb{Q}^n\) to the integral points \(L(\mathbb{Z})\) of a rational \(k\)-dimensional subspace \(L\in{\mathrm{Gr}}_{n,k}(\mathbb{Q})\) in the projective Grassmannian variety \({\mathrm{Gr}}_{n,k}\) of \(k\)-dimensional subspaces of \(\mathbb{R}^n\) represented in a basis is an integral quadratic form in \(k\) variables that is well defined up to a change of basis. That is, \(L\) has an associated `shape' \([L(\mathbb{Z})]\) in the space \(S_k\) of positive-definite real quadratic forms in \(k\) variables up to similarity, which may be identified with the homogeneous space \(\mathrm{O}_k(\mathbb{R})\backslash\mathrm{PGL}_k(\mathbb{R})/\mathrm{PGL}_k(\mathbb{Z})\) giving it a natural probability measure inherited from Haar measure. Similarly there is an associated shape \([L^{\perp}(\mathbb{Z})]\in S_{n-k}\), finally giving a triple \((L,[L(\mathbb{Z})],[L^{\perp}(\mathbb{Z})])\). Writing \({\mathrm{disc}}_Q(L)\) for the discriminant of \(Q\) restricted to \(L(\mathbb{Z})\), the problem studied here is find conditions that guarantee the equidistribution of \((L,[L(\mathbb{Z})],[L^{\perp}(\mathbb{Z})])\) as \({\mathrm{disc}}_Q(L)\to\infty\) in the space \({\mathrm{Gr}}_{n,k}(\mathbb{R}) \times S_k\times S_{n-k}\). The general conjecture is that if \(k\geqslant2\) and~\(n-k\geqslant2\) then there is equidistribution as the discriminant goes to infinity through values for which the set of subspaces \(L\) with that discriminant is non-empty (this is assumed throughout). Equidistribution of just the second two components has been extensively studied with various restrictions, including effective versions in certain dimensions. The triple equidistribution problem at hand was settled for \(k=2\) and \(n-k=2\) under a congruence condition when \(Q\) is the sum of four squares by stronger results of \textit{M. Einsiedler} with the first and third author [Duke Math. J. 171, No. 7, 1469--1529 (2022; Zbl 1503.37008)]; this was extended to arbitrary quadratic forms by the first and third author [in preparation]. Here the remaining cases are resolved, once again under some congruence conditions when \((k,n)\neq(2,4)\). The approach is to first prove relevant results on the equidistribution of certain adelic orbits in homogeneous dynamics. The simultaneous shape equidistribution results are then deduced from these.