Classification of $(\widetilde{Sp}(n,\mathbb{R})\times\widetilde{Sp}(1,\mathbb{R}))$-Manifolds

Abstract: Let $M$ be an analytic complete finite volume pseudo-Riemannian manifold and $\widetilde{Sp}(n,\mathbb{R})\times\widetilde{Sp}(1,\mathbb{R})$ a connected semisimple Lie group such that its Lie algebra is $\mathfrak{sp}(n,\mathbb{R})\oplus\mathfrak{sp}(1,\mathbb{R})$. We characterize the structure of the manifold $M$ assuming that the Lie group $\widetilde{Sp}(n,\mathbb{R})\times\widetilde{Sp}(1,\mathbb{R})$ acts isometrically on $M$ and that its dimension satisfies $3+n(2n+1)<\dim(M)\leq(n+1)(2n+3)$.