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Central characters for smooth irreducible modular representations of \(\operatorname{GL}_2(\mathbb Q_p)\). (English) Zbl 1266.22018

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 L. Barthel and 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 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.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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References:

[1] L. BARTHEL - R. LIVNEÂ, Irreducible modular representations of GL2 of a local field, Duke Math. J., 75, no. 2 (1994), pp. 261-292. · Zbl 0826.22019 · doi:10.1215/S0012-7094-94-07508-X
[2] L. BARTHEL - R. LIVNEÂ, Modular representations of GL2 of a local field: the ordinary, unramified case, J. NumberTheory, 55, no. 1 (1995), pp. 1-27. · Zbl 0841.11026 · doi:10.1006/jnth.1995.1124
[3] I. BERNSÏTEIÆN - A. ZELEVINSKIIÆ, Representations of the group GL(n; F), where F is a local non-Archimedean field, Russian Math. Surveys, 31, no. 3 (1976), pp. 1-68.
[4] C. BREUIL, Sur quelques repreÂsentations modulaires et p-adiques de GL2(Qp): I, Compositio Math., 138, no. 2, (2003), pp. 165-188.
[5] V. PASÏKUÅNAS, The image of Colmez’s MontreÂal functor, preprint, 2010.
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