Torsion theories for Iwahori-Hecke modules in characteristic $p$

By work of Bernstein and Borel the smooth representation theory of a $p$-adic reductive group $G$ with characteristic zero coefficients is very closely related to the module theory of the corresponding Iwahori-Hecke algebra $H$. With characteristic $p$ coefficients this relation is much more complicated and still mysterious. I will introduce a canonical torsion pair in the module category of $H$ (but which depends on $G$) whose associated class of torsion free modules allows a natural fully faithful embedding into the category of smooth mod $p$ representations of $G$. This is work in progress with R. Ollivier.