  We study invariant probability measures on the homogeneous space
$\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ for the action of subgroups
of $\mathrm{SL}_n(\mathbb R)$ of the form $SF$ where $F$ is generated by one
parameter unipotent groups and $S$ is a one parameter $\mathbb
R$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains
an element with only one eigenvalue less than one (counted with multiplicity)
and others bigger than one we prove that all the $SF$ invariant and ergodic
probability measures on $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ are
homogeneous.
