  Denote by $SL_n(\mathbb R)$ the group of $n\times n$ real matrices with
determinant one, $A$ the subgroup consisting of the diagonal matrices with
positive entries, and $SL_n(\mathbb R)/A$ the manifold of left cosets $gA$,
$g\in SL_n(\mathbb R)$. In this paper, we will be concerned with the harmonic
analysis on the homogeneous space $SL_n(\mathbb R)/A$ when $n=3$. In
particular, we provide explicit generators and their relations for the algebra
of the invariant differential operators. Then we prove that some of the
non-central generators are essentially self-adjoint.
