A partial Cayley transform for Siegel-Jacobi disk

Abstract. Let Hg and Dg be the Siegel upper half plane and the gen-eralized unit disk of degree g respectively. Let C(h,g) be the Euclidean space of all h × g complex matrices. We present a partial Cayley trans-form of the Siegel–Jacobi disk Dg × C(h,g) onto the Siegel–Jacobi space Hg × C(h,g) which gives a partial bounded realization of Hg × C(h,g) by Dg × C(h,g). We prove that the natural actions of the Jacobi group on Dg×C(h,g) and Hg×C(h,g) are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differ-ential operators on the Siegel–Jacobi disk Dg×C(h,g) invariant under the natural action of the Jacobi group on Dg × C(h,g) explicitly