Rodier type theorem for generalized principal series

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