Invariant Metrics and Laplacians on Siegel-Jacobi Disk

Let $ \mathbb{D}_n $ \mathbb{D}_n be the generalized unit disk of degree n. In this paper, Riemannian metrics on the Siegel-Jacobi disk $ \mathbb{D}_n $ \mathbb{D}_n × ?^(m,n) which are invariant under the natural action of the Jacobi group are found explicitly and the Laplacians of these invariant metrics are computed explicitly. These are expressed in terms of the trace form.