Smooth representations modulo \(\ell\) of \(\mathrm{GL}_m(D)\). (Représentations lisses modulo \(\ell\) de \(\mathrm{GL}_m(D)\).) (English) Zbl 1293.22005
Let F be a non-Archimedean locally compact field of residue characteristic \(p\), let D be a finite-dimensional central division F-algebra, and let R be an algebraically closed field of characteristic different from \(p\). The authors classify all smooth irreducible representations of GL\(_m\)(D) for \(m \geq 1,\) with coefficients in R, in terms of multisegments, generalizing works by Zelevinski, Tadić, and Vignéras. They prove that any irreducible R-representation of GL\(_m\)(D) has a unique supercuspidal support and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal support, and one by aperiodic multisegments, classifying representations with a given cuspidal support. These constructions are made in a purely local way, with a substantial use of type theory.
Reviewer: L. N. Vaserstein (University Park)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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