Weighted Bergman spaces associated with causal symmetric spaces
Let \(X=H\backslash G\) be an irreducible symmetric space, \({\mathbf g}={\mathbf h}+{\mathbf q}\) the corresponding decomposition of the Lie algebra \({\mathbf g}\) of \(G\), and \({\mathbf g}={\mathbf k}+{\mathbf p}\) a compatible Cartan decomposition. Then \(X\) is called compactly causal if there is a non-zero vector \(v\in V={\mathbf q}\cap {\mathbf k}\) which commutes with all of \(V\). This is equivalent to the assumption that \(L^2(X)\) admits a holomorphic discrete series representation. It implies that \(H_{{\mathbf C}}\backslash G_{{\mathbf C}}\) contains certain proper \(G\)-invariant Stein domains \(D\) arising from certain convex subsets in \(i{\mathbf q}\). The article under review investigates the Bergman space \(B^2(D)\) of holomorphic \(L^2\)-functions on such domains and calculates a Plancherel measure for this \(G\)-Hilbert space \(B^2(D)\). This result implies in particular that \(B^2(D)\neq\{0\}\). Part of the motivation for these investigations comes from the ``Gelfand Gindikin program'' [see \textit{I. M. Gel'fand} and \textit{S. G. Gindikin}, Funct. Anal. Appl. 11, 258--265 (1978); translation from Funkts. Anal. Prilozh. 11, No. 4, 19--27 (1977; Zbl 0449.22018), and \textit{G. I. Ol'shanskii}, Differ. Geom. Appl. 1, No. 3, 235--246 (1991; Zbl 0789.22011)]. Note: In the introduction a reference [HiO196] is quoted, which is missing from the references. Apparently this is the book of \textit{J. Hilgert} and \textit{G. Ólafsson}, Causal symmetric spaces. Geometry and harmonic analysis. San Diego, CA: Academic Press (1977; Zbl 0931.53004).
scientific article; zbMATH DE number 1334549
scientific article