}}=\bigsqcup_{\delta \text{ strong real form of } G^{\Gamma}}\Pi^{z}(G(\mathbb{R},\delta))^{mic}_{\psi_{\mathbf{a}}}$$ where $$\begin{aligned}     \Pi^{z}(G(\mathbb{R},\delta))^{mic}_{\psi_{\mathbf{a}}}=\{\pi\otimes\chi(\mathbf{a},\delta):\pi\in \Pi^{z}(G(\mathbb{R},\delta))^{mic}_{\