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15 May 2015 Ordinary representations of G ( Q p ) and fundamental algebraic representations
Christophe Breuil, Florian Herzig
Duke Math. J. 164(7): 1271-1352 (15 May 2015). DOI: 10.1215/00127094-2916104


Let G be a split connected reductive algebraic group over Q p such that both G and its dual group G ˆ have connected centers. Motivated by a hypothetical p -adic Langlands correspondence for G ( Q p ) , we associate to an n -dimensional ordinary (i.e., Borel-valued) representation ρ : Gal ( Q ¯ p / Q p ) G ˆ ( E ) a unitary Banach space representation Π ( ρ ) ord of G ( Q p ) over E that is built out of principal series representations. (Here, E is a finite extension of Q p .) Our construction is inspired by the “ordinary part” of the tensor product of all fundamental algebraic representations of G . There is an analogous construction over a finite extension of F p . When G = G L n , we show under suitable hypotheses that Π ( ρ ) ord occurs in the ρ -part of the cohomology of a compact unitary group.


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Christophe Breuil. Florian Herzig. "Ordinary representations of G ( Q p ) and fundamental algebraic representations." Duke Math. J. 164 (7) 1271 - 1352, 15 May 2015.


Received: 20 June 2012; Revised: 3 June 2014; Published: 15 May 2015
First available in Project Euclid: 14 May 2015

zbMATH: 1321.22019
MathSciNet: MR3347316
Digital Object Identifier: 10.1215/00127094-2916104

Primary: 22E50
Secondary: 11F70 , 11F80

Keywords: $p$-adic Langlands , fundamental algebraic representations , Galois representations , mod $p$ Langlands , representations of $p$-adic groups

Rights: Copyright © 2015 Duke University Press


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Vol.164 • No. 7 • 15 May 2015
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