Journal of the
Korean Mathematical Society JKMS

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

Keywords: invariants, invariant differential operators, the Minkowski-Euclid space

MSC numbers: Primary 13A50, 32Wxx, 15A72