The Capelli Identity, Tube Domains, and the Generalized Laplace Transform*

The key result in this paper is a proof of a large class of identities, generalizing the Capelli identity [Cl. The Capelli identity is a centerpiece of 19th century invariant theory. It asserts the equality of two differential operators on the n2-dimensional space M(n, R) of n x n real matrices. Let E, denote the ijth elementary matrix, let xii denote the linear functions on M(n, R) dual to E,, and write 3, for the partial differential operator a/ax,. Then this identity says