Invariant Differential Operators on the Minkowski-Euclid Space

Details Invariant Differential Operators on the Minkowski-Euclid Space Invariant Differential Operators on the Minkowski-Euclid Space

2006-11-13

arxiv.org/abs/math/0611389v5

Mathematics

Differential Geometry

revised version : more new results and more open problems. page 3 : Correction of the Killing form and the inner product page

Scientific paper

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).

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