New infinite families of $q$-analogs of group divisible designs with arbitrary block dimension

This paper is mainly devoted to constructions of $q$-analogs of group divisible designs and their applications. We give a complete description of the action of $G=\text{GL}(m,q^l)$ on ${\mathrm{\Omega }}_k^{k-1}$, where $3\le k\le \min \{m+1,l\}$ and ${\mathrm{\Omega }}_k^{k-1}$ is the set of $k$-subspaces of $\text{GF}(q{)}^{ml}$ whose $\text{GF}(q^l)$-span has dimension $k-1$. We do this by relating the $G$-orbits on ${\mathrm{\Omega }}_k^{k-1}$ to the corresponding Singer cycle orbits on subspaces of $\text{GF}(q{)}^l$. From the properties of the $G$-incidence matrix between $2$-subspaces and $k$-subspaces, we obtain plenty of new infinite families of simple $q$-analogs of group divisible designs with arbitrary block dimension. We further establish a recursive construction for simple $q$-analogs of pairwise balanced designs and then produce new infinite families of such designs. We also obtain plenty of infinite families of non-simple subspace $2$-designs through the above two types of designs.