Eisenstein integral

In mathematical representation theory, the Eisenstein integral is an integral introduced by in the representation theory of semisimple Lie groups, analogous to Eisenstein series in the theory of automorphic forms.
 used Eisenstein integrals to decompose the regular representation of a semisimple Lie group into representations induced from parabolic subgroups. 
 gave a survey of Harish-Chandra's work on this.
Definition.
 defined the Eisenstein integral by
where: