On double cosets of groups $GL(n)$ with respect to subgroups of block strictly triangular matrices

We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.