#### Vol. 4, No. 6, 2010

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

### Rachel Ollivier

Vol. 4 (2010), No. 6, 701–742
##### Abstract

We classify the simple supersingular modules for the pro-$p$-Iwahori Hecke algebra $\mathsc{ℋ}$ 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 $\mathsc{ℋ}$-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 $\mathsc{ℋ}$-modules in light of this process.

##### Keywords
mod $p$ representations of Hecke algebras and $p$-adic groups, parabolic induction, integral Bernstein presentation, integral Satake transform
##### Mathematical Subject Classification 2000
Primary: 20C08
Secondary: 20G05, 22E50