On differential operators attached to certain representations of classical groups (Q762536)

Let \(\sigma\) : GL\({}_ m({\mathbb{C}})\to GL_ n({\mathbb{C}})\) be an irreducible representation which occurs in the space of all homogeneous polynomial functions of degree r on the complex symmetric matrices. To each such \(\sigma\) and another representation \(\zeta\) of \(GL_ m({\mathbb{C}})\) the author attaches a certain non-holomorphic differential operator \(\Delta_{\zeta,\sigma}\) which sends a scalar-valued holomorphic Eisenstein series to a non-holomorphic vector valued one. Exact criteria for the non-vanishing of the latter can be obtained by studying the operator \(\Delta_{\zeta,\sigma}\). This problem is solved in this paper not only for \(GL_ m({\mathbb{C}})\), but for the automorphism groups of the classical Hermitian symmetric spaces.