Irreducible smooth representations in defining characteristic without central character
Abstract
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 \mathbb {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 ) \operatorname {GL}_n(F) over E E without a central character. We also construct irreducible smooth representations of GL n ( F ) \operatorname {GL}_n(F) over E E with simultaneously a central character, nonscalar endomorphisms, and if n > 3 n>3 , without a Hecke eigenvalue.
Fields of Study & Subjects
Advanced Algebra and Geometry
Finite Group Theory Research
Algebraic structures and combinatorial models
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 3
- Pages
- 1325–1337
- Publication date
- 16 January 2026
- Language
- EN