Branching laws of Klein four-symmetric pairs for \(\textrm{Sp}(n,\mathbb {R})\)

For the real symplectic groups \(G=\textrm{Sp}(n,\mathbb {R})\) $G=\text{Sp}(n,\mathbb{R})$ , we classify all the Klein four-symmetric pairs \((G,G^\Gamma )\) $(G,G^{\mathrm{\Gamma }})$ , and determine whether there exist infinite-dimensional irreducible \((\mathfrak {g},K)\) $(\mathfrak{g},K)$ -modules discretely decomposable upon restriction to \(G^\Gamma \) $G^{\mathrm{\Gamma }}$ . As a consequence, we obtain a similar result to Chen and He (Int J Math 34(1):2250094, 2023, Corollary 21).