On double cosets of groups GL (<i>n</i>) with respect to subgroups of block strictly triangular matrices

We parametrize the space of double cosets group GL(n,k) with respect to two subgroups T−, T+ block strictly triangular matrices. In appendix, we consider quasi-regular representation GL(n,C) in L2 on T−∖GL(n,C), observe that it admits an additional symmetries, find joint spectrum, and is multiplicity free.