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We classify non-polar irreducible representations of connected compact Liegroups whose orbit space is isometric to that of a representation of a finiteextension of $Sp(1)^k$ for some $k>0$. It follows that they are obtained fromisotropy representations of certain quaternion-K\"ahler symmetric spaces byrestricting to the "non-$Sp(1)$-factor".

Representations with $Sp(1)^k$-reductions and quaternion-K\"ahler symmetric spaces