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