Discrete decomposable branching laws and proper momentum maps (Q2888819)

Let \(G\) be a Lie group with Lie algebra \(\mathfrak{g}\), \(H\) a subgroup of \(G\) with Lie algebra \(\mathfrak{h}\) and \(\mathcal{O}\) a coadjoint orbit. Let \(\mu:\mathcal{O}\longrightarrow I\mathfrak{h}^\ast\) be the moment map corresponding to the Hamiltonian action of \(H\) on \(\mathcal{O}\) and \(\pi_{\mathcal{O}}\) the irreducible unitary representation corresponding to \(\mathcal{O}\) by means of the Kirillov map \(\mathfrak{g}^\ast/G\longrightarrow \widehat{G}\). Suppose that \(\mathfrak g\) is simple with Cartan decomposition \(\mathfrak{g}=\mathfrak{k}+\mathfrak{p}\) and set \(C_{\mathfrak{k}}^\ast=([\mathfrak{k},\mathfrak{k}]+\mathfrak{p})^\bot\). The author proves that if \(\mathfrak{g}\) is simple Hermitian, \(\mathcal{O}\) is any nonzero coadjoint orbit such that \(\mathcal{O}\cap iC_{\mathfrak{k}}^\ast\neq\emptyset\) and \((\mathfrak{g},\mathfrak{h})\) is a symmetric pair, then the map \(\mu\) is proper iff \(\pi_{\mathcal{O}}|_H\) decomposes discretely.