sheaf ${\widetilde{\mathfrak{E}}_{\vec{k}}}$. As a consequence of eq. , the morphism ${\widetilde{\Theta}_{\vec{k}}}$ is ${G_{\vec{k}}}$-equivariant, namely, the following diagram commutes $$\xymatrix@C+1em{
l_{12}^*{\widetilde{\mathfrak{E}}_{\vec{k},\infty}}\ar[r]^-{l_{12}^*{\widetilde{\Theta}_{\vec{k}}}} \ar[d]_{\Psi_\infty} & {\mathcal{O}}^{\oplus r}_{{{\ell_\infty}\times{P_{\vec{k}}}}\times{G_{\vec{k}}}}  \\
{\rho_{\infty}}^*{\widetilde{\mathfrak{E}}_{\vec{k},\infty}}\ar[ru]_-{{\rho_{\infty}}^*{\widetilde{\Theta}_{\vec{k}}}}  
}
\label{eqtTh*}$$ where $\Psi_\infty=\Psi|_{{{\ell_\infty}\times{P_{\vec{k}}}}\times{G_{\vec{k}}}}$ and $l_{12}:{{\ell_\infty}\times{P_{\vec{k}}}}\times{G_{\vec{k}}}\longrightarrow{{\ell_\infty}\times{P_{\vec{k}}}}$ is the projection.

The moduli space ${{\mathcal{M}}^n(r,a,c)}$
===========================================

In this section we give the moduli space ${{\mathcal{M}}^n(r,a,c)}$ a scheme structure, and prove the first part of the Main Theorem. The space ${{\mathcal{M}}^n(r,a,c)}$ can be set-theoretically identified with the quotient $\left.{P_{\vec{k}}}\right/{G_{\vec{k}}}$. We denote by ${\pi}:{P_{\vec{k}}}\longrightarrow{{\mathcal{M}}^n(r,a,c)}$ the natural projection.

\[proMk\] The orbit space ${{\mathcal{M}}^n(r,a,c)}$ is a smooth algebraic variety, and ${P_{\vec{k}}}$ is a locally trivial principal ${G_{\vec{k}}}$-bundle over it.

In order to prove this Theorem, we need to investigate some properties of the ${G_{\vec{k}}}$-action on ${P_{\vec{k}}}$.

\[lm2\] Let $\mathcal{E}$ and $\mathcal{E}'$ be sheaves on $\Sigma_n$ that are torsion-free and trivial at infinity. There is an injection $$\xymatrix{0 \ar[r] &
\text{\emph{Hom}}(\mathcal{E},\mathcal{E}') \ar[r]^-{R} & \text{\emph{Hom}}(\mathcal{E}|_{\ell_{\infty}},\mathcal{E}'|_{\ell_{\infty}})\simeq\text{\emph{End}}(\mathbb{C}^r)
}
\label{eq380}$$ where $R$ is the restriction morphism.

If $\mathcal{E}$ and $\mathcal{E}'$ are locally free, one has ${\operatorname{Hom}}\left({\mathcal{E}},{\mathcal{E}}'\right)\simeq H^0({\mathcal{E}}^*\otimes{\mathcal{E}}')$. The sheaf ${\mathcal{E}}^*\otimes{\mathcal{E}}'$ is locally free and trivial at infinity, so that the result follows by twisting the structure sequence of ${\ell_\infty}$ by it and taking cohomology (see Lemma \[lm3\]). In general, one can conclude because the morphism 