Locally analytic vectors of unitary principal series of \(\mathrm{GL}_2(\mathbb Q_p)\)
Authors' introduction: ``Let \(F\) be a finite extension of \(\mathbb Q_p\). The aim on the \(p\)-adic local Langlands programme initiated by Breuil is to look for a `natural' correspondence between certain \(n\)-dimensional \(p\)-adic representations of \(\text{GL}(\overline{\mathbb Q}_p/F)\) and certain Banach space representations of \(\text{GL}_n(F)\). Thanks to much recent work, especially that of Colmez and Paskunas, we have gained a fairly clear picture in the case \(F=\mathbb Q_p\) and \(n=2\) which is the so-called \(p\)-adic local Langlands correspondence for \(\text{GL}_2(\mathbb Q_p)\) established a functorial bijection between two-dimensional irreducible \(p\)-adic representations of \(\text{GL}(\overline{\mathbb Q}_p/\mathbb Q_p)\) and non-ordinary irreducible admissible unitary representations of \(\text{GL}_2(\mathbb Q_p)\).''NEWLINENEWLINE Locally analytic vectors of unitary principal series representations of \(\text{GL}_2(\mathbb Q_p)\) play an important role in the \(p\)-adic local Langlands correspondence -- they correspond to trianguline representations of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\). Let \({\mathcal S}_{\text{irr}}\) be the parametrizing space of two-dimensional irreducible trianguline representations of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\). For \(s\in{\mathcal S}_{\text{irr}}\), let \(V(s)\) be the corresponding trianguline representation, and let \(\Pi(V(s))\) be the admissible unitary representation of \(\text{GL}_2(\mathbb Q_p)\) attached to \(V(s)\) by the \(p\)-adic local Langlands correspondence.NEWLINENEWLINE ``\textit{M. Emerton} [Pure Appl. Math. Q. 2, No. 2, 279--393 (2006; Zbl 1254.11106)] made a conjectural description of the subspace of locally analytic vectors \(\Pi(V)_{\mathrm{an}}\) for all unitary principal series \(\Pi(V)\)'' (Conjecture 1.1).NEWLINENEWLINE NEWLINEIn this paper, the authors prove Emerton's conjecture for \(p>2\) and \(V(s)\) non-exceptional. The proof builds on \textit{P. Colmez}'s machinery of \(p\)-adic local Langlands correspondence for \(\text{GL}_2(\mathbb Q_p)\) developed in [Paris: Société Mathématique de France, Astérisque 330, 281--509 (2010; Zbl 1218.11107)].
scientific article; zbMATH DE number 6037744