Model categories and pro-$p$ Iwahori-Hecke modules

Let $G$ denote a possibly discrete topological group admitting an open subgroup $I$ which is pro-$p$. If $H$ denotes the corresponding Hecke algebra over a field $k$ of characteristic $p$ then we study the adjunction between $H$-modules and $k$-linear smooth $G$-representations in terms of various model structures. If $H$ is a Gorenstein ring we single out a full subcategory of smooth $G$-representations which is equivalent to the category of all Gorenstein projective $H$-modules via the functor of $I$-invariants. This applies to groups of rational points of split connected reductive groups over finite and over non-archimedean local fields, thus generalizing a theorem of Cabanes. Moreover, we show that the Gorenstein projective model structure on the category of $H$-modules admits a right transfer. On the homotopy level the right derived functor of $I$-invariants then admits a right inverse and becomes an equivalence when restricted to a suitable subcategory.