##### Matthias Strauch, November 3, 2023

In the first part of the talk we will describe a classification of 2-dimensional \(p\)-adic representations of the absolute Galois group of Qp, followed by a review of Colmez’ \(p\)-adic Langlands correspondence which associates to such a representation r an admissible unitary \(p\)-adic Banach space representation \(\Pi(r)\) of \(G=GL_2(Q_p)\).

Then we consider the question for which r the associated locally analytic representation \(\Pi(r)^\text{la}\) is generated by \(H\)-analytic vectors for certain analytic subgroups \(H\) of \(G\).

We then state a conjecture which says that \(\Pi(r)^\text{la}\) is generated by its \(H\)-analytic vectors if and only if the representation is \(B\)-admissible for a certain period ring related to \(H\). Finally we give some evidence for the conjecture.