Block decomposition of the category of smooth \(\ell\)-modular representations of finite length of \(\mathrm{GL}_m(D)\). (Décomposition en blocs de la catégorie des représentations \(\ell\)-modulaires lisses de longueur finie de \(\mathrm{GL}_m(D)\).) (French. English summary) Zbl 1535.22055
Summary: Let \(\mathbb{F}\) be a non-Archimedean locally compact field of residue characteristic \(p\), let \(G\) be an inner form of \(\mathrm{GL}_n(\mathbb{F})\) with \(n\geq 1\) and \(\ell\) be a prime number different from \(p\). We describe the block decomposition of the category of smooth representations of finite length of \(G\) with coefficients in \(\bar{\mathbb{F}}_{\ell}\). Unlike the case of complex representations of an arbitrary \(p\)-adic reductive group and that of \(\ell\)-modular representations of \(\mathrm{GL}_n(\mathbb{F})\), several non-isomorphic supercuspidal supports may correspond to the same block. We describe the (finitely many) supercuspidal supports corresponding to a given block. We also prove that a supercuspidal block is equivalent to the principal (that is, the one which contains the trivial character) block of the multiplicative group of a suitable division algebra, and we determine those irreducible representations having a nontrivial extension with a given supercuspidal representation of \(G\).
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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