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arXiv · 1206.4413

Ordinary representations of G(Q_p) and fundamental algebraic representations

Christophe Breuil·Florian Herzig
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Abstract

Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence for G(Q_p) we associate to an n-dimensional ordinary (i.e. Borel valued) representation rho : Gal(Q_p-bar/Q_p) to G-hat(E) a unitary Banach space representation Pi(rho)^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. In the latter case, when G=GL_n we show under suitable hypotheses that Pi(rho)^ord occurs in the rho-part of the cohomology of a compact unitary group. We also prove a weaker version of this result in the p-adic case.

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Christophe Breuil, Florian Herzig. 2012-06-20. Ordinary representations of G(Q_p) and fundamental algebraic representations. https://doi.org/10.1215/00127094-2916104

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