I started to read Colin Bushnell's notes on this title. The last theorem in the 3rd section claims that if $\pi$ is an irr. smooth rep. of $GL_N(F)$ containing a non-split fundamental stratum, then $\pi$ contains a simple stratum.

The first step in the proof is to show that one can assume that the non-split fundamental stratum $[\mathfrak{a},n,n-1,b]$ contained in $\pi$ is such that $e(\mathfrak{a})$ and $n$ are relatively prime.

I think that there is a mistake in this proof: The proof in the notes.

The containment of the unit subgroups is seem to be (clearly) the other way around and not as mentioned.

Is there a work around or other way to prove it?

Thanks, Zahi.