A note on certain Iwahori--Hecke modules of $GL_3$ in characteristic $p$
In this note, we show that the analogue of certain finiteness result of Barthel--Livn$\acute{\text{e}}$ on $GL_2$ fails for $GL_3$. More precisely, for the pro-$p$ Iwahori invariants of the maximal compact induction of $GL_3$ with trivial coefficient, we show there exist non-zero Iwahori--Hecke submodules of \emph{infinite} codimension.