Unitary highest weight modules
Let G be one of the groups \(SU(p,q)\), \(SO^*(2n)\), or \(\tilde Sp(2n,{\mathbb{R}})\). The problem is to associate the unitary highest weight modules of G with the coadjoint orbits of G and, hence, tie in with the usual picture of the unitary dual of G. Some standard notation: let K be a maximal compact subgroup of G, let \({\mathfrak G}\) and \({\mathfrak K}\) be the complexified Lie algebras of G and K, respectively, and let \(\theta\) be the Cartan involution. Denote by \({\mathfrak l}+{\mathfrak u}^ a \theta\)-stable parabolic subalgebra of \({\mathfrak G}\) and let L be the stabilizer of \({\mathfrak l}\) in G. Finally, let \(\Pi\) be an irreducible unitary representation of L. Considering \(\Pi\) as a representaion of \({\mathfrak l}\) we can write \(\Pi =(\lambda,\pi)\) where \(\lambda\) is in the dual of the center of \({\mathfrak l}\) and \(\pi =\Pi |_{[{\mathfrak l,l}]} \). Using the data \(\lambda,\pi\), the author defines a module (a derived functor module) \(A(\lambda,\pi)\). The association between \(A(\lambda,\pi)\) and a coadjoint orbit of G is given for the cases where \(\pi\) is trivial or \(\pi\) is a unipotent representation of L; according to the main theorem, these are the only cases needed. Theorem. Under certain stated conditions on \(\lambda\) and \(\pi\), \(A(\lambda,\pi)\) is an irreducible unitary holomorphic representation of G. Any irreducible unitary holomorphic representation of G is equivalent to some \(A(\lambda,\pi)\) with \(\lambda,\pi\) satisfying the conditions. Note: The \(A(\lambda,\pi)\) are not all distinct.
scientific article; zbMATH DE number 4010813
scientific article