Two questions

Question the first

Let ${\mathcal{O}}$ be the integer ring of a mixed characteristic local field ${F/\mathbf{Q}_{p}}$. Fix an algebraic closure ${\overline{F}}$ with absolute Galois group ${\mathrm{Gal}(\overline{F}/F)}$, and let ${T}$ be a Galois-stable ${\mathbf{Z}_{p}}$-lattice in a crystalline representation of ${\mathrm{Gal}(\overline{F}/F)}$ all of whose Hodge-Tate weights are ${0}$ and ${-1}$. By a remarkable theorem of Breuil and Kisin, the association ${G\rightsquigarrow T_{p}G}$ is an equivalence of categories between ${p}$-divisible groups over ${\mathcal{O}}$ and lattices in crystalline representations of the above type.

Let us say ${T}$ as above is ${F}$analytic if the following two conditions hold:

1. There is an ${\mathcal{O}}$-module structure on ${T}$ for which the Galois action is ${\mathcal{O}}$-linear.

2. For all ${\sigma\in\mathrm{Hom}(F,\overline{F})}$ except for our initial embedding ${F\subset\overline{F}}$, the map

$\displaystyle \left(T\otimes_{\mathcal{O},\sigma}\widehat{\overline{F}}\right)^{\mathrm{Gal}(\overline{F}/F)}\otimes_{F}\widehat{\overline{F}}\rightarrow T\otimes_{\mathcal{O},\sigma}\widehat{\overline{F}}$

is an isomorphism.

Is is true that ${T}$ is ${F}$-analytic if and only if the associated ${p}$-divisible group ${G}$ admits a strict ${\mathcal{O}}$-action? (Recall that a strict ${\mathcal{O}}$-action is a ring map ${\mathcal{O}\rightarrow\mathrm{End}(G)}$ such that the induced ${\mathcal{O}}$-action on ${\mathrm{Lie}(G)}$ agrees with the one given by the structure map ${G\rightarrow\mathrm{Spec}\mathcal{O}}$.)

Question the second

Let ${F/\mathbf{Q}_p}$ be a mixed characteristic local field with uniformizer ${\varpi}$, ${G=\mathrm{GL}_{n}(F)}$, ${B}$ the lower triangular Borel, ${T\cong(F^{\times})^{n}}$ the diagonal maximal torus. Fix another local field ${L/\mathbf{Q}_p}$, and let ${\delta:T\rightarrow L^{\times}}$ be a continuous character, identified with a tuple of characters ${\delta_{i}:F^{\times}\rightarrow L^{\times}}$ in the obvious way. Suppose for all ${1\leq i\leq n}$ we have the inequality ${\sum_{1\leq j\leq i}v(\delta_{j}(\varpi))\geq0}$, with equality for ${i=n}$. Does the locally analytic induction

$\displaystyle \Pi(\delta)=\mathrm{Ind}_{B}^{G}(\delta)^{\mathrm{an}}$

admit a ${G}$-invariant norm?