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Rodier type theorem for generalized principal series

Journal article, 2021

Given a regular supercuspidal representation ρ 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(IndPG(ρ)) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation IndPG(ρ), 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 = N (M) / M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group W = N (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 and SO , 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. M G T G 2 n 2 n + 1

Regular supercuspidal representation

Generalized principal series

Generic

Discrete series

Tempered representation