Irreducible smooth representations in defining characteristic without central character

Let p > 3 p>3, n > 1 n>1 an integer, and F F be a non-archimedean local field with residue field a proper finite extension of F p Fₚ. Let E E be an algebraically closed countable field extension of the residue field of F F. In this short note, we explain how the methods from Le Math. Res. Lett. 26 (2019), 1747–1758 and Ghate, Le, and Sheth Represent. Theory 27 (2023), 1088–1101 can be used to construct irreducible smooth representations of GL n ⁡ (F) GLₙ (F) over E E without a central character. We also construct irreducible smooth representations of GL n ⁡ (F) GLₙ (F) over E E with simultaneously a central character, nonscalar endomorphisms, and if n > 3 n>3, without a Hecke eigenvalue.