Rodier type theorem for generalized principal series

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--H{\"o}lder 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. Indeed, such a beautiful structure theorem also holds for finite central covering groups.