Let $A$ be a finite dimensional algebra over a field $k$, let $Y$ be a f.g. $A$-module, let $A-\mathbf {mod}_Y$ be the full subscategory of endomorphic images of powers of $Y$ in $A-\mathbf {mod}$. We show that if $\mathrm {End}_AY$ is self-injective then $A-\mathbf {mod}_Y$ and $(\mathrm {End}_AY)^\circ -\mathbf {mod}$ are equivalent. The subcategory $A-\mathbf {mod}_Y$ is not stable by quotients in $A-\mathbf {mod}$ but when it contains the simple modules this leads to a criterion of complete reducibility. This applies to $A=kG$, $Y=\mathrm {Ind}_U^Gk$ when $G$ is a finite BN-pair of characteristic $p=\mathrm {char}(k)$ and $U$ is one of its Sylow $p$-subgroups ; $\mathrm {End}_{kG}Y$ is a modular Hecke algebra. The equivalence of categories and the criterion of complete redfucibility are applied to certain questions on $kG$-modules stemming from the study of finite geometries and extensions of simple modules.