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.