Mod $p$ representations of $p$-adic ${\rm GL}_n$ and its pro-$p$ Hecke algebra

by Rachel Ollivier, Université de Versailles-Saint Quentin.

The aim of this talk is to explore the possibility of a mod $p$ analog for the theories of complex representations of $p$-adic reductive groups and of Hecke algebras presented in Shaun Stevens' course.
I will focus on the case of ${\rm GL}_n$.
I will give results about the related structures of the (pro-$p$) Iwahori-Hecke algebra and the spherical Hecke algebra with integer/mod $p$ c\oe fficients and will discuss the properties of the natural link between pro-$p$ Hecke modules and smooth mod $p$ representations of $p$-adic ${\rm GL}_n$.