({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda))= \left[\begin{array}{cc} \alpha_{0,k}(z) {\operatorname{Id}}&0\\ 0& \alpha_{0,k}(z){\operatorname{Id}}\end{array}\right],$$ where ${\operatorname{Id}}\colon ({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G) \to ({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)$.  Define 