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...
\begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon) V_{\beta
k}][G_{\beta\nu}^r(\epsilon) k}]\\
&[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon) V_{\beta k}]
\\
&[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon) V_{\beta
k}][G_{\beta\nu}^>(\epsilon) k}]\\
&[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon) V_{\beta k}]
\end{align*}
Inserting the expressions for the self-energies we get
\begin{align*}
...
&+g_{k}^{h,>}(\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\epsilon)V_{\beta q}^* g_{q}^{h,a}(\epsilon) ]\,,
\end{align*}
Then we get
\begin{eqnarray}
&&M^>(\omega)=\frac{2 \begin{align*}
&M^>(\omega)=\frac{2 i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta,\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{
V_{\beta k}^{*} g_{k}^{h,r}(\epsilon) V_{\alpha k} G^{r}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,>}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu}
G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))+ G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\\
\nonumber
&&
V_{\beta &+{\beta k}^{*} g_{k}^{h,r}(\epsilon) V_{\alpha k} G^{>}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,<}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu}
G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))\\ \nonumber
&&
V_{\beta G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\\
&+V_{\beta k}^{*} g_{k}^{h,>}(\epsilon) V_{\alpha k} G^{a}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,a}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\omega+\epsilon) (f_e(\omega+\epsilon)+f_h(\epsilon))\Biggr\}
\end{eqnarray} \end{align*}
We now use the explicit expressions for the self-energies
\begin{eqnarray}
&&M^>(\omega)=\frac{2i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{ [\Sigma^{r,h}_{0,\beta\alpha}(\epsilon) G^{r}_{\alpha\delta}(\epsilon) \Sigma^{h,>}_{0,\delta\gamma} G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)) +
