Joint Equidistribution of Subspaces of Bounded Height in Rigid Adelic Spaces

For every $1\le d<n$, we prove joint equidistribution, as the height tends to infinity, of $d$-dimensional $K$-subspaces in a rigid adelic space over a number field $K$, together with their archimedean Grassmannian images and the $K$-linear isometry classes of the normalized subspace and quotient. We identify the leading constant, prove no escape of mass from either normalized factor, and obtain global equidistribution against bounded continuous test functions. Over a general number field, the two normalized shapes are coupled by a determinant-class relation; conditional on the determinant class, their limiting law is the product of the Haar-induced probability measures on the corresponding fibers.