The analysis of irreducibility in the class of elementary representations of a complex semisimple Lie group

We study the class of “elementary” representations for a complex semisimple Lie group, obtained by analytic continuation from the Gel'fand–Na\u{i}mark “fundamental series”. We establish necessary and sufficient conditions for the irreducibility of these representations. Here the term “irreducibility” is to be understood to mean both topological irreducibility and complete irreducibility in the sense of R. Godement.