On two step tensor modules of the maximal compact subgroups of inner type noncompact real simple Lie groups (Q1890467)

Let \(\Gamma_K\) be the set of all \(P_K\)-dominant integral forms on \(b_C\). The main results of the paper are: (i) Assume that \(\mu \in \Gamma_K\) is admissible. Then the multiplicity \(M(\mu)\) of \(V_{\mu}\) in the \(K\)-module \(M(\mu)\) is given by \(M(\mu) = \#\{\omega \in \Sigma_n:\omega\) is \(P_{K(\mu)}\)-highest\(\}\). (ii) Let \(\mu \in \Gamma_K\). Then there exists a standard triple \((P_K,P(\Theta),P)\) such that \(\mu \in C(\Theta)^*\). Moreover, we have \(K(\Theta) = K(\mu)\).