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The Sp(1)-Kepler Problems

Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-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 $\widetilde {\mathrm O}^*(4n)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm {O}^*}(4n)$ with highest weight $$(\underbrace{-1, ..., -1}_{2n-1}, -(1+\barσ)),$$ which occurs at the right-most nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. Here $\barσ$ is the highest weight of $σ$. Furthermore, it is shown that the correspondence $σ\leftrightarrow \mathscr H(σ)$ is the theta-correspondence for dual pair $(\mathrm{Sp}(1), \mathrm{O}^*(4n))\subseteq\mathrm{Sp}_{8n}(\mathbb R)$.