Cayley transform and generalized Whittaker models for irreducible highest weight modules (Q2740890)

The purpose of the present article is to describe the generalized Whittaker models for irreducible admissible highest weight \((g,K)\)-modules \(L (\tau)\) for a connected simple Lie group \(G\) of Hermitian type (where \(K\) is a maximal subgroup of \(G\) and \(g\) is the Lie algebra of \(G\)) by using certain invariant differential operators \(D_{\tau^*}\) of gradient type on the Hermitian symmetric space \(K \backslash G\), associated to the \(K\)-representation \(\tau^*\), dual to \(\tau\). This operator \(D_{\tau^*}\) was introduced by [\textit{M. G. Davidson, T. J. Enright} and \textit{R. J. Stanke}, Math. Ann. 288, No. 4, 731-739 (1990; Zbl 0723.22017)]. \(L (\tau)\) embeds, with nonzero and finite multiplicity, into the generalized Gelfand-Graev representation \(\Gamma_{m(\tau)}\) associated to the unique open orbit in the associated variety of \(L (\tau)\). If \(L(\tau)\) is unitarizable, the author describes the space of infinitesimal homomorphisms from \(L(\tau)\) into \(\Gamma_{m(\tau)}\) in terms of the principal symbol at the origin of the differential operator \(D_{\tau^*}\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.22001].