AN INTEGRAL TRANSFORM AND UNITARY HIGHEST WEIGHT MODULES

Abstract

The unitary highest weight modules for $G={\rm U}(1,q)$, which occur as irreducible subrepresentations of the oscillator representation on a Fock space ${\mathcal F}$, can each be realized on a space of polynomial-valued functions over the bounded realization ${\bf B}^q$ of $G/K$. This is achieved via an integral transform constructed by L.~Mantini. A decomposition of these representations into $K$-types is given, including an explicit description of how Mantini's transform behaves on $K$-types. An inverse is produced for the transform, thus giving unitary structures for the geometric realizations of the unitary highest-weight modules over $G/K$. 1991 Mathematics Subject Classification: 22E45, 22E70.