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Let $p>2$ be a prime. Let $\rho$ be a crystalline representation of$G_{\mathbb{Q}_p}$ with distinct Hodge-Tate weights in $[0, p]$, such that itsreduction $\overline \rho$ is upper triangular. Under certain conditions, weprove that $\overline \rho$ has an upper triangular crystalline lift $\rho'$such that $\mathrm{HT}(\rho')=\mathrm{HT}(\rho)$. The method is based on theauthor's previous work, combined with an inspiration from the work ofBreuil-Herzig.

A note on crystalline liftings in the $\mathbb{Q}_p$ case