Irreducible smooth representations in defining characteristic without central character

Let $p>3$ and F be a non-archimedean local field with residue field a proper finite extension of Fₚ. Let E be an algebraically closed countable field extension of the residue field of F. In this short note, we explain how the methods from arXiv:1809.10247 and arXiv:2210.07281 can be used to construct irreducible smooth representations of GLₙ(F), $n>1$, over E without a central character.