The Capelli identity, tube domains, and the generalized Laplace transform (Q1814130)

Let \(\mathfrak n\) be a real (finite dimensional) simple and formally real Jordan algebra. For any \(x\in{\mathfrak n}\), let \(L(x)\) denote the operator of left multiplication by \(x\). Let \({\mathfrak p}=\{L(x)\mid x\in{\mathfrak n}\}\), \({\mathfrak t}=[{\mathfrak p,p}]\) and \({\mathfrak g}={\mathfrak t}+{\mathfrak p}\). Then \(\mathfrak g\) is a reductive Lie algebra. Let \(G\) (resp. \(K\)) be the analytic subgroup of \(GL({\mathfrak n})\) corresponding to \(\mathfrak g\) (resp. \(\mathfrak t\)). Then \(C=G/K\) is a symmetric space, which is a self-dual cone in \(\mathfrak n\). If one identifies \(\mathfrak n\) with its dual via a \(K\)- invariant positive definite bilinear form, then the norm \(\varphi\) of \(\mathfrak n\) (which is a polynomial function of degree \(n=\text{rk }{\mathfrak n}\) on \(\mathfrak n\)) corresponds to a differential operator \(\partial(\varphi)\) on \(\mathfrak n\), and for each positive integer \(m\), one may consider the so called ``generalized Cayley operator'' \(D_ m=\varphi^ m\partial(\varphi)^ m\). Since \(C\) is open, one may regard \(D_ m\) as a differential operator on \(C\). Let \(G=NAK\) be the Iwasawa decomposition of \(G\). Then the submanifold \(A\cdot e\) of \(C=G/K\) is transversal to each \(N\)-orbit. Identifying \(A\cdot e\) with \(A\), one has that the \(N\)-radial part of \(D_ m\) is a differential operator on \(A\). This corresponds, via the exponential map, to a polynomial function on \({\mathfrak a}^*\), where \(\mathfrak a\) is the Lie algebra of \(A\). This polynomial, after a standard \(\rho\)-shift, becomes a \(W\)-invariant polynomial \(p_ m\), where \(W\) is the Weyl group of \(\mathfrak a\) in \(K\). The main result of the paper is an explicit formula for \(p_ m\). This leads to some differential identities. When the cone of \(n\times n\) positive definite Hermitian matrices is considered, a particular choice of the ingredients of these identities gives precisely the classical Capelli identity \(\text{det}(\pi_{ij}+(n- i)\delta_{ij})=\text{det}(x_{ij})\text{det}(\partial_{ij})\), (where \(\pi_{ij}=\sum^ n_{k=1}x_{ki}\partial_{kj})\) of two differential operators on the \(n^ 2\)-dimensional space of \(n\times n\) real matrices.