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Ordinary parts of admissible representations of \(p\)-adic reductive groups. I: Definition and first properties. (English) Zbl 1205.22013

Berger, Laurent (ed.) et al., Représentations \(p\)-adiques de groupes \(p\)-adiques III: Méthodes globales et géométriques. Paris: Société Mathématique de France (ISBN 978-2-85629-282-2/pbk). Astérisque 331, 355-402 (2010).
Let \(p\) be a prime integer. Let \(G\) be a connected reductive \(p\)-adic group, \(P\) a parabolic subgroup of \(G\), \(\bar P\) an opposite parabolic to \(P\) and \(M=P \cap \bar P\) the corresponding Levi factor of \(P\) and \(\bar P\). Let \(k\) be a finite field of characteristic \(p\), and let \(\text{Mod}_{G}^{\text{adm}}(k)\) denote the category of admissible smooth \(G\) representations over \(k\). Parabolic induction yields a functor \(\text{Ind}_{\bar P}^{G}:\text{Mod}_{M}^{\text{adm}}(k)\to \text{Mod}_{G}^{\text{adm}}(k)\). A functor \(\text{Ord}_P:\text{Mod}_{G}^{\text{adm}}(k)\to \text{Mod}_{M}^{\text{adm}}(k)\) which is right adjoint to \(\text{Ind}_{\bar P}^{G}\) is called the functor of ordinary parts of admissible representations of \(G\) associated to \(P\).
The author of the paper under review studies the basic properties of this functor and presents some applications to the computation of Ext spaces in the category \(\text{Mod}_{G}^{\text{adm}}(k)\) when \(G=\text{GL}_2(\mathbb Q_{p})\). These computations play a role in the construction of the modular and \(p\)-adic local Langlands correspondences.
The paper under review is not just over the field \(k\), but over general Artinian local rings or even complete local rings having residue field \(k\). This approach develops the foundations of the theory of admissible representations over such coefficient rings. The paper is well-written and is an excellent source for researchers in the subject.
For the entire collection see [Zbl 1192.11002].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
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