The Sp(1)-Kepler Problems

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mathrm{S}\mathrm{p}(1)$, an $\mathrm{S}\mathrm{p}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension five. The dynamical symmetry group of this system is ${\tilde{\mathrm{O}}}^{\ast }(4n)$ with the Hilbert space of bound states $\mathcal{H}(\sigma )$ being the unitary highest weight representation of $\widetilde{{\mathrm{O}}^{\ast }}(4n)$ with highest weight $$(\underset{2n-1}{\underbrace{-1,\mathrm{.}\mathrm{.}\mathrm{.},-1}},-(1+\bar{\sigma })),$$ which occurs at the right-most nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. Here $\bar{\sigma }$ is the highest weight of $\sigma$. Furthermore, it is shown that the correspondence $\sigma \leftrightarrow \mathcal{H}(\sigma )$ is the theta-correspondence for dual pair $($\mathrm{S}\mathrm{p}$(1), $\mathrm{O}$^*(4n))\subseteq$\mathrm{S}\mathrm{p}$_{8n}(\mathbb R)$.