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From étale \(P_+\)-representations to \(G\)-equivariant sheaves on \(G/P\). (English) Zbl 1360.11131

Diamond, Fred (ed.) et al., Automorphic forms and Galois representations. Proceedings of the 94th London Mathematical Society (LMS) – EPSRC Durham symposium, Durham, UK, July 18–28, 2011. Volume 2. Cambridge: Cambridge University Press (ISBN 978-1-107-69363-0/pbk; 978-1-107-29752-4/ebook). London Mathematical Society Lecture Note Series 415, 248-366 (2014).
Summary: Let \(K/\mathbb Q_{p}\) be a finite extension with ring of integers \(o\), let \(G\) be a connected reductive split \(\mathbb Q_{p}\)-group of Borel subgroup \(P=TN\) and let \(\alpha\) be a simple root of \(T\) in \(N\). We associate to a finitely generated module \(D\) over the Fontaine ring over \(o \) endowed with a semilinear étale action of the monoid \(T_{+} \) (acting on the Fontaine ring via \(\alpha\)), a \(G(\mathbb Q_{p})\)-equivariant sheaf of \(o\)-modules on the compact space \(G(\mathbb Q_{p})/P(\mathbb Q_{p})\). Our construction generalizes the representation \(D\boxtimes \mathbb P^{1} \) of \( GL(2,\mathbb Q_{p})\) associated by P. Colmez [in: Représentations \(p\)-adiques de groupes \(p\)-adiques II: Représentations de \(\text{GL}_2 (\mathbb Q_p)\) et \((\varphi, \Gamma)\)-modules. Paris: Société Mathématique de France. 61–153 (2010; Zbl 1235.11107)] to a \((\varphi,\Gamma)\)-module \(D\) endowed with a character of \(\mathbb Q_{p}^{*}\).
For the entire collection see [Zbl 1310.11003].

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
11F85 \(p\)-adic theory, local fields
11F80 Galois representations
22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 1235.11107
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