Determination of the intertwining operators for holomorphically induced representations of Hermitian symmetric pairs
Let (G,K) be an irreducible Hermitian symmetric pair, let \({\mathfrak g}\) (resp. \({\mathfrak m})\) be the complexified Lie algebra of G (resp. of K), let \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}\) and \({\mathfrak m}\), and let \({\mathfrak W}^ m\) be the set of minimal length left coset representations. The authors show that \(Hom(N_ y,N_ x)\) is either C or O, and give the result in terms of the highest weights. Here, \(N_ z\) \((z=y\) or \(z=x)\) denotes the maximal \({\mathfrak m}\)-locally finite quotient of the \({\mathfrak g}\)-Verma module of highest weight \(z\rho\)-\(\rho\) with \(z\in {\mathfrak W}^{{\mathfrak m}}\) and \(\rho =1/2\sum_{\alpha \in \Delta^+}\alpha\), where \(\Delta^+\) is the set of positive \({\mathfrak h}\)-roots of \({\mathfrak g}\) [\textit{T. J. Enright} and \textit{B. Shelton}, Mem. Am. Math. Soc. 367 (1987; Zbl 0621.17004)].
scientific article; zbMATH DE number 4085931