Let G be the group of rational points of a split connected reductive group over a non-archimedean local field of residue characteristic p, and let \({\mathcal {H}}\) denote the pro-p Iwahori–Hecke algebra of G over a field of characteristic p. We study the parabolic induction functor for \({\mathcal {H}}\) -modules in terms of the Gorenstein projective model structures introduced by Hovey. Let \({\operatorname {Ho}}({\mathcal {H}})\) denote the associated homotopy category of this model structure. We show that \({\operatorname {Ho}}({\mathcal {H}})\) and its thick subcategory generated by the essential images of finitely many parabolic induction functors are related via a recollement of triangulated categories. We then investigate the isomorphism classes of simple \({\mathcal {H}}\) -modules in \({\operatorname {Ho}}({\mathcal {H}})\) and give a complete classification for simple supersingulars when \(G=\textrm{GL}_n\) .
