k}{\rm -cr}}$ if and only if the representation ${\mathcal{G}}_K\rightarrow {\mathrm{GL}}_n(L')$ defined by $x$ is crystalline with labelled Hodge-Tate weights $(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$. That this ring exists is the main result of [@Kisindef]. We write ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ for the rigid analytic space associated to ${\mathrm{Spf}}\, R_{\overline r}^{\square,{\bf k}{\rm -cr}}$. By [@Kisindef], it is smooth over $L$.

Let $\tilde r:{\mathcal G}_K\rightarrow {\mathrm{GL}}_n(R_{\overline r}^{\square,{\bf k}{\rm -cr}})$ be the corresponding universal deformation. By [@Kisindef Th.2.5.5] or [@BergerColmez Cor.6.3.3] there is a coherent $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$-module ${\mathcal{D}}$ that is locally on ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ free over $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$ together with a $\varphi\otimes {\mathrm{id}}$-linear automorphism $\Phi_{\rm cris}$ such that: $$({\mathcal{D}},\Phi_{\rm cris})\otimes_{{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}} k(x)\cong D_{\rm cris}\big(\tilde r\otimes_{R_{\overline r}^{\square,{\bf k}{\rm -cr}}} k(x)\big)$$ for all $x\in {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$. Fixing an embedding $\tau_0:K_0\hookrightarrow L$ we can define the associated family of Weil-Deligne representations: $$({\rm WD}(\tilde r),\Phi):=\big({\mathcal{D}}\otimes_{K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}},\tau_0\otimes{\mathrm{id}}}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}},\Phi_{\rm cris}^{[K_0:{\mathbb{Q}}_p]}\otimes{\mathrm{id}}\big)$$ on ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ whose isomorphism class does not depend on the choice of the embedding $\tau_0$.

Let $T^{\mathrm{rig}}\cong ({\mathbb G}_{{\rm m}}^{\mathrm{rig}})^{n}$ be the rigid analytic space over ${\mathbb{Q}}_p$ associated to the diagonal torus $T\subset {\mathrm{GL}}_{n}$ and let ${\mathcal{S}}_n$ be the Weyl group of 