  We obtain an irreducibility criterion for generalized principal series,
extending known and frequently employed results for principal series. Our
approach rests on a newly observed semi-direct product decomposition of the
relative Weyl group and its action on generalized principal series, in
conjunction with the theory of Jacquet module. We thus circumvent the obstacle
that the Weyl group of a general Levi subgroup is not a Coxeter group. Our
statements on irreducibility are formulated in terms of a subgroup of the
Knapp--Stein R-group, which arises naturally from our decomposition of the Weyl
group. The novel subgroup allows us to state a conjecture on general parabolic
induction of arbitrary irreducible admissible representations as opposed to
generalized principal series, which are associated with supercuspidal ones.
