Highest weight vectors for the principal series of semisimple Lie groups and embeddings of highest weight modules (Q909044)

Let G be a connected simple Lie group with finite center and K a maximal compact subgroup of G. Let \({\mathfrak g}\) denote the Lie algebra of G and \({\mathfrak g}_ C\) denote its complexification. By Casselman's subrepresentation theorem, every irreducible admissible (\({\mathfrak g}_ C,K)\)-module can be embedded into some member of the principal series induced from a minimal parabolic subgroup of G. The explicit description of embeddings of all the irreducible highest weight representations into the principal series is given by the method of highest weight vectors [the author, Adv. Stud. Pure Math. 14, 31-121 (1988)] in this paper, such a description may be derived from [\textit{N. R. Wallach}, Mém. Soc. Math. Fr., Nouv. Sér. 15, 291-305 (1984; Zbl 0556.22006)], the method given in this paper is different from Wallach's and more direct.