Parabolic induction and Hecke modules in characteristic $p$ for $p$-adic GL$_n$

Abstract

We classify the simple supersingular modules for the pro-$p$-Iwahori Hecke algebra $\mathcal{\mathscr{H}}$ of $p$-adic ${GL}_n$ by proving a conjecture by Vignéras about a mod $p$ numerical Langlands correspondence on the side of the Hecke modules. We define a process of induction for $\mathcal{\mathscr{H}}$-modules in characteristic $p$ that reflects the parabolic induction for representations of the $p$-adic general linear group and explore the semisimplification of the standard nonsupersingular $\mathcal{\mathscr{H}}$-modules in light of this process.