Muller–Speh–Vogan type irreducibility criterion for generalized principal series

Abstract

We provide an elementary way to obtain an irreducibility criterion for generalized principal series of p-adic reductive groups and the finite central covering groups, extending known and frequently employed results for principal series. Basically our approach is a counting argument which rests on a 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 relative Weyl group of a general Levi subgroup is not a Coxeter group. Along the way, we observe a structural irreducibility criterion with respect to Levi subgroups. Indeed, following Speh–Vogan's argument for real groups which is essentially a corollary of intertwining operator theory, the former result can be generalized to parabolic inductions inducing from essentially tempered representations, but the latter one can't be obtained in this way. In view of the lack of intertwining operator theory for mod ℓ representations, our counting argument may play a role there, as opposed to the failure of Speh–Vogan's argument.