Central characters for smooth irreducible modular representations of \(\operatorname{GL}_2(\mathbb Q_p)\) (Q1947818)

It is proved that every smooth irreducible \(\overline{ \mathbb F}_p\)-linear representation of \(\text{GL}_2(\mathbb Q_p)\) admits a central character. The proof is based on results of \textit{L. Barthel} and \textit{R. Livné} [Duke Math. J. 75, No. 2, 261--292 (1994; Zbl 0826.22019); J. Number Theory 55, No. 1, 1--27 (1995; Zbl 0841.11026)] and a result of \textit{C. Breuil} [Compos. Math. 138, No. 2, 165--188 (2003; Zbl 1044.11041)] on the classification of smooth irreducible \(\mathbb F_p\)-linear representations of \(\operatorname{GL}_2(\mathbb Q_p)\) admitting a central character. As a corollary, it is deduced that every smooth irreducible \(\overline{\mathbb F}_p\)-linear representation of \(\operatorname{GL}_2(\mathbb Q_p)\) is admissible, hence satisfies Schur's lemma.