Let \({\mathcal D}\) be the set of complex \(n\times n\)-matrices \(w\) such that \(I_n-ww^*\) is positive definite and \({\mathcal H}^2_k ({\mathcal D})\) be the space of holomorphic functions on \({\mathcal D}\) which are square integrable with respect to the measure \(\det(I_n- ww^*)dw\). In the survey article under review the author explains his results on the decomposition of \({\mathcal H}^2_k({\mathcal D})\) under the natural action of \(U(p,q)\times U(p,q)\) with \(n=p+q\). They are based on Howe duality and an intertwiner onto an appropriate Fock type Hilbert space.