Modulo $p$ parabolic induction of pro-$p$-Iwahori Hecke algebra

We study the structure of parabolic inductions of a pro-$p$-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo $p$ representations of pro-$p$-Iwahori Hecke algebra in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vigneras, it completes the classification of irreducible modulo $p$ representations.