Rigid analytic vectors of crystalline representations arising in\n $p$-adic Langlands

Let $\\mathbf{B}(V)$ be the admissible unitary\n$GL_2(\\mathbb{Q}_p)$-representation associated to two dimensional crystalline\nGalois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger\nand Breuil conjectured an explicit description of the locally analytic vectors\n$\\mathbf{B}(V)_{\\mathrm{la}}$ of $\\mathbf{B}(V)$ which is now proved by Liu.\nEmerton recently studied $p$-adic representations from the viewpoint of rigid\nanalytic geometry. In this article, we consider certain rigid analytic\nsubgroups of $GL(2)$ and give an explicit description of the rigid analytic\nvectors in $\\mathbf{B}(V)_{\\mathrm{la}}$. In particular, we show the existence\nof rigid analytic vectors inside $\\mathbf{B}(V)_{\\mathrm{la}}$ and prove that\nits non-null. This gives us a rigid analytic representation (in the sense of\nEmerton) lying inside the locally analytic representation\n$\\mathbf{B}(V)_{\\mathrm{la}}$.\n

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