An integral transform and ladder representations of \(U(p,q)\)
If \(G\) is a reductive Lie group, then a natural place to look for its representations is on spaces of sections of homogeneous bundles over \(G/K\), where \(K\) is the maximal compact subgroup of~\(G\). In particular, if the representation is unitarisable, then one might expect to be able to write down the unitary structure as some invariant construction on~\(G/K\). In this article, the author provides such a construction for the ladder representations of \({\text{U}}(p,q)\) by inverting an integral transform of \textit{L.Mantini} [J. Funct. Anal. 60, 211-241 (1985; Zbl 0565.22012)]. More specifically, he interprets Mantini's transform as mapping the Fock space realisation of the positive spin ladder representations to sections of an appropriate homogeneous holomorphic vector bundle over \({\text{U}}(p,q)/({\text{U}}(p)\times{\text{U}}(q))\). By finding an explicit inverse to this transform, he is able to pull back the standard unitary structure on the Fock space. The result is an integral against a \(G\)-invariant measure on \(G/K\) and, as such, this new way of viewing the unitary structure is quite natural and can be divorced from its origin. This is a carefully written and detailed article.
scientific article; zbMATH DE number 1344264