An integral transform and unitary highest weight modules (Q2766360)

In this paper, two realizations of the unitary highest weight modules for \(G=U(1,q)\) are studied, where \(U(p,q)\) is a Lie subgroup of \(GL(p+q,\mathbb{C})\). These modules occur as subrepresentations of the oscillator representation and its tensor powers on a Fock space \(\mathcal{F}\). On the other hand, they may be realized as a space of polynomial-valued functions over the bounded model \(\mathbb{B}^q\) of \(G/K\), where \(\mathbb{B}^q\) is simply the unit ball in \(\mathbb{C}^q\). The latter realization is obtained from the former via an intertwining integral transform developed by \textit{L. A. Mantini} [J. Funct. Anal. 60, 211-242 (1985; Zbl 0565.22012); Trans. Am. Math. Soc. 323, 583-603 (1991; Zbl 0747.22009)]. NEWLINENEWLINENEWLINEThe main aim of this paper is to describe the behavior of Mantini's transform at the \(K\)-type level, and then to produce an inversion formula. NEWLINENEWLINENEWLINEThe structure of the paper is as follows. The oscillator representation is introduced and Mantini's transform is presented. The groundwork for the inversion formula is given. The modules are then decomposed into \(K\)-types, and highest weight vectors are constructed for each \(K\)-type. An explicit description of Mantini's transform at the \(K\)-type level is produced and the norms of highest weight vectors are computed. Finally, the results are used to construct an inverse for Mantini's transform, thus producing unitary structures for the unitary highest weight modules for \(G=U(1,q)\) occurring over \(G/K\).