Weighted Bergman spaces and the holomorphic discrete series of \(\widetilde{U(p,q)}\)
The symmetric space \(SU(n,n)/S(U(p,q) \times U(p,q))\) with \(n= p+q\) can be realized as a domain \({\mathcal D}_{p,q}\) in the complex \(n\times n\) matrices, defined by the condition that \(I_{p,q}-w* I_{p,q}w\) is positive definite, where \(I_{p,q}\) is a diagonal matrix with \(p\) entries 1 and then \(q\) entries \(-1\). Applying a Cayley transform to \({\mathcal D}_{p,q}\) one obtains a different realization \(T_\Omega\setminus \Sigma_{p,q}\). Here \(\Omega\) is the cone of hermitian positive definite \(n\times n\) matrices, \(T_\Omega\) is the tube domain \(V+i\Omega\), where \(V\) is the space of all hermitian \(n\times n\) matrices, and \(\Sigma_{p,q}\) is the complex hypersurface in \(T_\Omega\) defined by the equation \(\det(z+iI_{p,q}) =0\). The author gives an explicit construction of a covering \(\widetilde {{\mathcal D}_{p,q}}\) of \({\mathcal D}_{p,q}\) with infinitely many sheets and considers a family of weighted Bergman spaces \({\mathcal H}^2_\alpha(\widetilde {{\mathcal D}_{p,q}})\) on this covering. The weight functions indexed by \(\alpha\) are the powers \(\det(I_{p,q}- w^*I_{p,q} w)^{\alpha-2n}\). His first main result is an explicit isomorphism between \({\mathcal H}^2_\alpha (\widetilde {{\mathcal D}_{p,q}})\) and the classical weighted Bergman spaces \({\mathcal H}^2_\alpha (T_\Omega)\) with respect to the weight functions \(\det(\text{Im} z)^\alpha\). The weighted Bergman spaces carry a natural representation of the universal covering group \(\widetilde {U(n,n)}\) which decomposes as a direct sum of highest weight representations. The author gives an explicit description of this decomposition and the corresponding expansion of the reproducing kernel of the weighted Bergman spaces in terms of holomorphic discrete series characters.
scientific article; zbMATH DE number 1462830