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 F_p . 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 ) GL_n(F) over E E without a central character. We also construct irreducible smooth representations of GL n ⁡ ( F ) GL_n(F) over E E with simultaneously a central character, nonscalar endomorphisms, and if n > 3 n>3 , without a Hecke eigenvalue.