at infinity for ${\mathcal{E}}_{\alpha,\beta}$, namely, a point $\theta\in{P_{\vec{k}}}$ is an isomorphism $\theta:{\mathcal{E}}_{\alpha,\beta,\infty}\stackrel{\sim}{\longrightarrow}{\mathcal{O}}_{{\ell_\infty}}^{\oplus r}$, where $(\alpha,\beta)=\tau(\theta)$.

-   One can lift $\rho_0$ to an action $\rho\colon{P_{\vec{k}}}\times{G_{\vec{k}}}\to{P_{\vec{k}}}$ by letting $$\rho(\theta,\bar\psi) = \theta \circ \Lambda_\infty\left(\alpha,\beta;{\bar{\psi}}\right)^{-1}\,,
    \label{eqGonP}$$ where $(\alpha,\beta)=\tau(\theta)$, and after letting $\left(\alpha',\beta'\right)={\bar{\psi}}\cdot(\alpha,\beta)$, the isomorphism $$\label{lambda}\Lambda_\infty\left(\alpha,\beta;{\bar{\psi}}\right):{\mathcal{E}}_{\alpha,\beta,\infty}\longrightarrow{\mathcal{E}}_{\alpha',\beta',\infty}$$ is induced by ${\bar{\psi}}:M(\alpha,\beta)\longrightarrow M\left(\alpha',\beta'\right)$. The identity $$\Lambda_\infty(\alpha,\beta,\bar\psi'\cdot\bar\psi) = \Lambda_\infty(\alpha',\beta',\bar\psi') \circ
    \Lambda_\infty(\alpha,\beta,\bar\psi)$$ ensures that $\rho$ is indeed an action. We note that the projection $\tau\colon {P_{\vec{k}}}\to{L_{\vec{k}}}$ becomes a ${G_{\vec{k}}}$-equivariant morphism.

\[pagcond2\]

We have now all ingredients needed to state our main result.

\[teo2\] The set ${\mathcal{M}}^n(r,a,c)$ is nonempty if and only if $c+\frac{1}{2}na(a-1)\geq0$. If this is the case, it can be given a structure of smooth algebraic variety of dimension $2rc+(r-1)na^2$ by representing it as the quotient ${\mathcal{M}}^n(r,a,c)={P_{\vec{k}}}/{G_{\vec{k}}}$. Moreover, ${\mathcal{M}}^n(r,a,c)$ turns out to be a fine moduli space of framed sheaves on $\Sigma_n$.

Note that $$\dim {\mathcal{M}}^n(r,a,c) = 2rc+(r-1)na^2 = 2r\Delta,$$ where $\Delta = c_2 - \frac{r-1}{2r}c_1^2$ is the discriminant of the sheaves parametrized by $ {\mathcal{M}}^n(r,a,c)$.

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

One knows by instanton counting that ${{\mathcal{M}}^n(r,a,c)}$ is connected [@BPT], hence, being smooth, it is irreducible as well.

Families of framed sheaves
==========================

In this section we explain how the varieties ${L_{\vec{k}}}$ and ${P_{\vec{k}}}$ arise and 