Ordinary representations of \(G(\mathbb{Q}_p)\) and fundamental algebraic representations. (English) Zbl 1321.22019

Breuil, Christophe; Herzig, Florian

Duke Math. J. 164, No. 7, 1271-1352 (2015).

From the paper: “The \(p\)-adic Langlands program for the group \(\text{GL}_2(\mathbb{Q}_p)\) is now well understood, from both local and global points of view (see [Proceed. Intern. Congress of Math., II, Hindustan Book Agency, New Delhi, 203–230 (2010)] for an overview). In particular, to (essentially) any continuous representation \(\rho: \text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\to\text{GL}_2(E)\) (where \(E\) is a finite extension of \(\mathbb{Q}_p\)) one can associate a unitary continuous representation \(\Pi(\rho)\) of \(\text{GL}_2(E)\) on a \(p\)-adic Banach space over \(E\). Likewise, to (essentially) any continuous representation \(\overline\rho: \text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\to \text{GL}2_(k_E)\) (where \(k_E\) is the residue field of \(E\)), one can associate a smooth representation \(\Pi(\overline\rho)\) of \(\text{GL}_2(\mathbb{Q}_p)\) over \(k_E\). It is moreover expected that the \(p\)-adic Langlands correspondence – if there is one – for any other group beyond \(\text{GL}_2(\mathbb{Q}_p)\) (e.g. \(\text{GL}_2(L)\) or \(\text{GL}_3(\mathbb{Q}_p)\) or \(\text{GSp}_4(\mathbb{Q}_p)\)) will be significantly more involved than that for \(\text{GL}_2(\mathbb{Q}_p)\). The aim of the present work is nevertheless to start to investigate the possible shape of the representation(s) \(\pi(\rho)\) and \(\Pi(\overline\rho)\) when \(\rho\), \(\overline\rho\) take values in split reductive groups other than \(\text{GL}_2\).”
“Let \(G\) be a split connected reductible algebraic group over \(\mathbb{Q}_p\) such that both \(G\) and its dual group \(G\) have connected centers. Motivated by a hypothetical \(p\)-adic Langlands correspondence for \(G(\mathbb{Q}_p)\), we associate to an \(n\)-dimensional ordinary (i.e. Borel-valued) representation \(\rho:\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\to\widehat G(E)\) a unitary Banach space representation \(\Pi(\rho)^{\text{ord}}\) of \(G(\mathbb{Q}_p)\) over \(E\) that is built out of principal series representations. 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 \(\mathbb{F}_p\). When \(G=\text{GL}\), we show under suitable hypotheses that \(\Pi(\rho)^{\text{ord}}\) occurs in the \(p\)-part of the cohomology of a compact unitary group.”

Reviewer: Andrzej Dąbrowski (Szczecin)

MSC:

22E50	Representations of Lie and linear algebraic groups over local fields
11F80	Galois representations
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

\(p\)-adic Langlands; \(\text{mod\,}p\) Langlands; Galois representations; representations of \(p\)-adic groups; fundamental algebraic representations; local-global compatibility

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