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?

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