Rigid analytic vectors of crystalline representations arising in \(p\)-adic Langlands (Q2162782)

The paper under review calculates the subspace of rigid analytic vectors of the unitary Banach representation \(B(V)\) of \(\mathrm{GL}_2(\mathbb{Q}_p)\) associated to a \(2\)-dimensional crystalline Galois representation \(V\) of the Galois group of \(\mathbb{Q}_p\) by the \(p\)-adic local Langlands correspondence of \(\mathrm{GL}_2(\mathbb{Q}_p)\). The subspace of locally analytic vectors of \(B(V)\) was previous known in terms of locally analytic principal series and locally algebraic representations, e.g., [\textit{R. Liu}, Compos. Math. 148, No. 1, 28--64 (2012; Zbl 1267.11059)]. The paper gives an explicit description of the analytic vectors for certain rigid analytic congruence subgroups of \(B(V)\) and shows that the subspaces are not zero.