Rodier type theorem for generalized principal series

Abstract

Given a regular supercuspidal representation rho of the Levi subgroup M of a standard parabolic subgroup P=MN in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set JH(Ind^G_P(rho )) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation Ind^G_P(rho ), vastly generalizing Rodier structure theorem for P=B=TU Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group W_M=N_G(M)/M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group W_T=N_G(T)/T is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split (G)Sp_{2n} and SO_{2n+1}, and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.