  For two positive integers m and n, we let ${\mathcal P}_n$ be the open convex
cone in ${\mathbb R}^{n(n+1)/2}$ consisting of positive definite n x n real
symmetric matrices and let ${\mathbb R}^{(m,n)}$ be the set of all m x n real
matrices. In this article, we investigate differential operators on the
non-reductive manifold ${\mathcal P}_n \times {\mathbb R}^{(m,n)}$ that are
invariant under the natural action of the semidirect product group GL(n,R)
$\ltimes {\mathbb R}^{(m,n)}$ on the Minkowski-Euclid space ${\mathcal P}_n
\times {\mathbb R}^{(m,n)}$. These invariant differential operators play an
important role in the theory of automorphic forms on GL(n,R) $\ltimes {\mathbb
R}^{(m,n)}$ generlaizing that of automorphic forms on GL(n,R).
