Abstract
Let n≥2 be a positive integer. To each irreducible representation σ of Sp(1), an Sp(1)-Kepler problem in dimension (4n−3) is constructed and analyzed. This system is superintegrable, and when n=2 it is equivalent to a generalized MICZ-Kepler problem in dimension of 5. The dynamical symmetry group of this system is Õ∗(4n) with the Hilbert space of bound states H(σ) being the unitary highest weight representation of O∗˜(4n) with highest weight, (−1,⋯,−1,−(1+σ¯)), which occurs at the rightmost nontrivial reduction point in the Enright–Howe–Wallach classification diagram for the unitary highest weight modules. Here σ¯ is the highest weight of σ. Furthermore, it is shown that the correspondence σ↔H(σ) is the theta-correspondence for dual pair (Sp(1),O∗(4n))⊆Sp(8n,R).
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