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We obtain an irreducibility criterion for generalized principal series,extending known and frequently employed results for principal series. Ourapproach rests on a newly observed semi-direct product decomposition of therelative Weyl group and its action on generalized principal series, inconjunction with the theory of Jacquet module. We thus circumvent the obstaclethat the Weyl group of a general Levi subgroup is not a Coxeter group. Ourstatements on irreducibility are formulated in terms of a subgroup of theKnapp--Stein R-group, which arises naturally from our decomposition of the Weylgroup. The novel subgroup allows us to state a conjecture on general parabolicinduction of arbitrary irreducible admissible representations as opposed togeneralized principal series, which are associated with supercuspidal ones.

Muller type irreducibility criterion for generalized principal series