Breuil's locally analytic vectors conjecture for crystabelian representations

Let $V$ be an irreducible crystabelian representation of $G_{{\mathbb {Q}}_p}$ with Hodge–Tate weights $(0,k-1)$ for an integer $k\geq 2$. Let $(\alpha,\beta)$ be the associated pair of smooth characters of ${\mathbb {Q}}_p^\times$, and let $A(\alpha)$ and $A(\beta)$ be the corresponding locally analytic principal series; let $\pi(\beta)$ be the associated locally algebraic representation. Write $A(\alpha)\oplus_{\pi(\beta)}A(\beta)$ for their amalgamated sum over $\pi(\beta)$. Breuil's conjecture. If $\alpha\neq\beta$, the natural continuous ${\rm GL}_2({\mathbb {Q}}_p)$-equivariant map

$$A(\alpha)\oplus_{\pi(\beta)}A(\beta)\longrightarrow\mathrm{B}(V)_{\mathrm{an}}$$

is a topological isomorphism.

This conjecture concerns an explicit description of locally analytic vectors in the $p$-adic local Langlands correspondence. The paper presents it as a conjecture of Breuil and proves the relevant description in the stated irreducible, crystabelian, Frobenius semi-simple case.