 ${\mathbb{V}}_{-\Delta}(\mu_j)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_j)^G$ is a trivial, even-dimensional $N(K)$-representation.  0.  There exist $\mu_i,\ \mu_j\in\sigma(-\Delta,\Omega)$ such that $m_0<\mu_i<M_0$