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Denote by $SL_n(\mathbb R)$ the group of $n\times n$ real matrices withdeterminant one, $A$ the subgroup consisting of the diagonal matrices withpositive 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 harmonicanalysis on the homogeneous space $SL_n(\mathbb R)/A$ when $n=3$. Inparticular, we provide explicit generators and their relations for the algebraof the invariant differential operators. Then we prove that some of thenon-central generators are essentially self-adjoint.

Harmonic analysis on the fourfold cover of the space of ordered triangles I: the invariant differentials