The Sp(1)-Kepler Problems

TL;DR

This paper introduces a family of super integrable $ ext{Sp}(1)$-Kepler problems in higher dimensions, analyzing their symmetry groups and representation theory, and connecting them to theta-correspondences in dual pairs.

Contribution

It constructs and analyzes $ ext{Sp}(1)$-Kepler problems in dimensions $(4n-3)$, revealing their super integrability, symmetry groups, and representation-theoretic properties, including a novel connection to theta-correspondences.

Findings

System is super integrable.

Dynamical symmetry group is $ ilde{O}^*(4n)$.

Hilbert space of bound states is a specific highest weight representation.

Abstract

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ 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 ${\mathscr H}(\sigma)$ being the unitary highest weight representation of $\widetilde {\mathrm {O}^*}(4n)$ with highest weight $$(\underbrace{-1, ..., -1}_{2n-1}, -(1+\bar\sigma)),$$ 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 $\sigma\leftrightarrow \mathscr H(\sigma)$ is the theta-correspondence for dual pair $(\mathrm{Sp}(1), \mathrm{O}^*(4n))\subseteq\mathrm{Sp}_{8n}(\mathbb R)$.