Irreducibles as kernels of intertwinings among principal series

To understand the larger context, first note that Casselman’s subrepresentation theorem asserts that any irreducible representation [1] π of G is a subrepresentation of some one of the principal series representations Is defined below. [2] Further, the quotient Is/π (or a further quotient that is irreducible) again imbeds in some Is′ with another parameter value s′. Thus, any irreducible π appears as a (possibly subrepresentation of a) kernel of a G-homomorphism Is → Is′ among principal series. Still further, examination of the eigenvalues of the center of the enveloping algebra [3] on the Is shows that the only values s′ such that Is′ has a non-trivial G-homorphism Is → Is′ are s′ = s and s′ = 1 − s. There is a natural integral for a G-homomorphism Is → Is′ , which can be evaluated in terms of the gamma function on adroitly chosen vectors in the principal series. But, again, the computation itself is understandable without necessarily fully appreciating this grounding of it.