INVARIANT DIFFERENTIAL OPERATORS ON THE MINKOWSKI-EUCLID SPACE

Abstract: For two positive integers m and n, we let

{m a t h c a l P}_{n}

be the open convex cone in

{m a t h b b R}^{n (n + 1) / 2}

consisting of positive definite n x n real symmetric matrices and let

{m a t h b b 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

{m a t h c a l P}_{n} i m e s {m a t h b b R}^{(m, n)}

that are invariant under the natural action of the semidirect product group GL(n,R)

l t i m e s {m a t h b b R}^{(m, n)}

on the Minkowski-Euclid space

{m a t h c a l P}_{n} i m e s {m a t h b b R}^{(m, n)}

. These invariant differential operators play an important role in the theory of automorphic forms on GL(n,R)

l t i m e s {m a t h b b R}^{(m, n)}

generlaizing that of automorphic forms on GL(n,R).