$({\mathrm{GL}}_{n},T)$ acting on $T$, and thus on $T^{\mathrm{rig}}$, in the usual way. Recall that the map: $${\mathrm{diag}}(\varphi_1,\varphi_2,\dots,\varphi_n)\mapsto {\rm coefficients\ of}\ (X-\varphi_1)(X-\varphi_2)\dots (X-\varphi_n)$$ induces an isomorphism of schemes over ${\mathbb{Q}}_p$: $$T/{\mathcal{S}}_n\buildrel\sim\over\longrightarrow {\mathbb G}_{{\rm a}}^{n-1}\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p} {\mathbb G}_{{\rm m}}$$ and also of the associated rigid analytic spaces. We deduce that the coefficients of the characteristic polynomial of the Frobenius $\Phi$ on ${\rm WD}(\tilde r)$ determine a morphism of rigid analytic spaces over $L$: $${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}\longrightarrow T_L^{{\mathrm{rig}}}/{\mathcal{S}}_n.$$ Let us define: $$\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}:={\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{T_L^{\mathrm{rig}}/{\mathcal{S}}_n}T_L^{\mathrm{rig}}.$$ Concretely $\widetilde{\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm-cr}$ parametrizes crystalline framed ${\mathcal{G}}_K$-deformations $r$ of $\overline r$ of labelled Hodge-Tate weights ${\bf k}$ together with an ordering $(\varphi_1,\dots,\varphi_n)$ of the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$.

\[reduced\] The rigid analytic space $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is reduced.

It is sufficient to prove this result locally. Let $\mathrm{Sp}\,C$ be an admissible irreducible affinoid open subspace of $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ whose image in $T_L^{\rm rig}/\mathcal{S}_n$ is contained in an admissible affinoid open irreducible subspace $\mathrm{Sp}\,A$ of $T_L^{\rm rig}/\mathcal{S}_n$. As both $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ and $T_L^{\rm rig}/\mathcal{S}_n$ are smooth over $L$ we can find an admissible open affinoid covering of $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ by such $\mathrm{Sp}\,C$. The map $T_L^{\rm rig}\rightarrow T_L^{\rm rig}/\mathcal{S}_n$ 