The Plancherel theorem for biinvariant Hilbert spaces (Q1974443)

Let \({\mathfrak g}\) be a finite-dimensional Lie algebra which admits an open cone \(W\) invariant under the adjacent action. Then for each Lie group \(G\) with Lie algebra \({\mathfrak g}\) one can build a semigroup \(\Gamma= G\text{ Exp}(iW)\) which is a complex manifold and has \(G\) as a group of units. The author studies Hilbert spaces of holomorphic functions on left and right \(G\)-invariant open subdomains of \(\Gamma\). He shows that the resulting unitary \(G\times G\)-representation decomposes as a multiplicity free direct integral of representations of the form \(\pi^*\otimes\pi\) where \(\pi\) is a unitary highest weight representation of \(G\).