  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.
