particular for any point $\theta\in{P_{\vec{k}}}$ one has the natural isomorphism $$\left({\widetilde{\mathfrak{E}}_{\vec{k}}}\right)_\theta\simeq{\mathcal{E}}_{\tau(\theta)}\,.
\label{eqtEi}$$ We shall call ${\widetilde{\mathfrak{E}}_{\vec{k},\infty}}$ the restriction of ${\widetilde{\mathfrak{E}}_{\vec{k}}}$ to ${{\ell_\infty}\times{P_{\vec{k}}}}$, so that ${\widetilde{\mathfrak{E}}_{\vec{k},\infty}}\simeq\left({\operatorname{id}}_{{\mathbb{P}^1}}\times\tau\right)^*{\breve{\mathfrak{E}}_{\vec{k},\infty}}\simeq{\tilde{\mathfrak{u}}}_2^*\tilde{\mathfrak N}_{\vec k}$.

The flatness of ${\widetilde{\mathfrak{E}}_{\vec{k}}}$ on ${P_{\vec{k}}}$ follows from the identification ${\Sigma_n\times{P_{\vec{k}}}}\simeq{P_{\vec{k}}}\times_{{L_{\vec{k}}}}{\Upsilon}$, while the isomorphism comes from [@Har Prop 9.3], as ${{\ell_\infty}\times{P_{\vec{k}}}}\simeq{P_{\vec{k}}}\times_{{L_{\vec{k}}}}{\Upsilon_\infty}$. The last statement is trivial.

One can extend the action $\rho$ of ${G_{\vec{k}}}$ on ${P_{\vec{k}}}$ to actions ${\tilde{\rho}}$ on ${\Sigma_n\times{P_{\vec{k}}}}$ and ${\rho_{\infty}}$ on ${{\ell_\infty}\times{P_{\vec{k}}}}$ by setting $$\left\{
\begin{aligned}
{\tilde{\rho}}&:={\operatorname{id}}_{\Sigma_n}\times\rho\\
{\rho_{\infty}}&:={\operatorname{id}}_{{\ell_\infty}}\times\rho\,.
\end{aligned}
\right.$$

\[proGontMo\] The sheaves ${\widetilde{\mathfrak{E}}_{\vec{k}}}$ and ${\widetilde{\mathfrak{E}}_{\vec{k},\infty}}$ are, respectively, isomorphic to the cohomologies of the monads $$\begin{aligned}
{\widetilde{\mathbb{M}}_{\vec{k}}}&:=\left({\operatorname{id}}_{\Sigma_n}\times\tau\right)^*{\breve{\mathbb{M}}_{\vec{k}}};\\
{\widetilde{\mathbb{M}}_{\vec{k},\infty}}&:=\left({\operatorname{id}}_{{\ell_\infty}}\times\tau\right)^*{\breve{\mathbb{M}}_{\vec{k},\infty}}.\end{aligned}$$ Both monads ${\widetilde{\mathbb{M}}_{\vec{k}}}$ and ${\widetilde{\mathbb{M}}_{\vec{k},\infty}}$ are ${G_{\vec{k}}}$-equivariant.

The sheaf ${\widetilde{\mathfrak{E}}_{\vec{k}}}$ admits a ${G_{\vec{k}}}$-linearization $\Psi$ satisfying the isomorphism $$+++\left(\Psi|_{{{\ell_\infty}\times{P_{\vec{k}}}}\times{G_{\vec{k}}}}\right)_{\left(\theta,{\bar{\psi}}\right)}\simeq\Lambda_\infty\left(\alpha,\beta;{\bar{\psi}}\right)$$ for any point $\left(\theta,{\bar{\psi}}\right)\in{P_{\vec{k}}}\times{G_{\vec{k}}}$.

One defines a morphism of monads $
m_{12}^*{\breve{\mathbb{M}}_{\vec{k}}}\to \left({\operatorname{id}}_{\Sigma_n}\times\rho_0\right)^*{\breve{\mathbb{M}}_{\vec{k}}}$ by letting $$\begin{array}{ccl}
(m_{12}^*{\breve{\mathbb{M}}_{\vec{k}}})_{(x,\alpha,\beta,{\bar{\psi}})}&\longrightarrow&( \left({\operatorname{id}}_{\Sigma_n}\times\rho_0\right)^*{\breve{\mathbb{M}}_{\vec{k}}})_{(x,\alpha,\beta,{\bar{\psi}})}\\[5pt]
(u,v,w) & \longmapsto& (\phi(u),\psi(v),\chi(w))\end{array}\,.$$ where $m_{12}\colon {\Upsilon}\times{G_{\vec{k}}}\to {\Upsilon}$ is the canonical projection and $\rho_0$ was defined in eq. . This induces by pullback a morphism $\Psi\colon m_{12}^\ast {\widetilde{\mathfrak{E}}_{\vec{k}}}\to \left({\operatorname{id}}_{\Sigma_n}\times\rho_0\right)^*{\widetilde{\mathfrak{E}}_{\vec{k}}}$ which satisfies the property in the statement of this Corollary.

Since ${P_{\vec{k}}}$ is the bundle of linear frames of ${\mathfrak{N}_{\vec{k}}}$, there exists a canonical isomorphism $\widetilde{\mathfrak{N}}_{\vec{k}}\longrightarrow{\mathcal{O}}_{{P_{\vec{k}}}}^{\oplus r}$, which can be regarded as a framing ${\widetilde{\Theta}_{\vec{k}}}$ for the 