\(p\)-modular representations of \(p\)-adic groups (Q2817984)

Fix two primes \(p\) and \(l\). The motivation for studying the representation theory of \(p\)-adic groups comes from the local Langlands conjectures.NEWLINENEWLINE In the case \(l\neq p\), we are in the setting of the classical local Langlands correspondence, which can be stated (roughly) as the following injective map (for any \(n\geq 1\)):NEWLINENEWLINE Continuous representation of \(\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\) on \(n\)-dimensional \(\overline{\mathbb{Q}_l}\)-vector spaces, up to isomorphism irreducible, admissible representations of \(\text{GL}_n(\mathbb{Q}_p)\) on \(\overline{\mathbb{Q}_l}\)-vector spaces, up to isomorphism.NEWLINENEWLINE The correspondence was first established by \textit{M. Harris} and \textit{R. Taylor} [The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)] and Henniart (2000), and more recently by Scholze.NEWLINENEWLINE In the case \(l=p\), we would like to have a \(p\)-adic analog of the above correspondence, to be dubbed the ``\(p\)-adic local Langlands correspondence''. Additionally, we would like to have a mod-\(p\) version.NEWLINENEWLINE When \(n=2\), such a correspondence has been made precise and proven by the work of Breuil, Colmez, Paškūnas, and others.NEWLINENEWLINE These notes are an introduction to the \(p\)-modular (or ``mod-\(p\)'') representation theory of \(p\)-adic reductive groups. We will focus on the group \(\mathrm{GL}_2(\mathbb Q_p)\), but we try to provide statements that generalize to an arbitrary \(p\)-adic reductive group \(G\) (for example, \(\mathrm{GL}_2(\mathbb Q_p)\)).NEWLINENEWLINEFor the entire collection see [Zbl 1343.20001].