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

## TL;DR

This paper classifies specific non-polar irreducible representations of compact Lie groups related to quaternion-K"ahler symmetric spaces, focusing on those with orbit spaces isometric to certain $Sp(1)^k$-extensions.

## Contribution

It provides a classification of non-polar irreducible representations linked to quaternion-K"ahler symmetric spaces, highlighting their relation to $Sp(1)^k$-extensions.

## Key findings

- Identifies all such non-polar irreducible representations.
- Shows they derive from isotropy representations of quaternion-K"ahler symmetric spaces.
- Clarifies the structure of orbit spaces in these cases.

## Abstract

We classify non-polar irreducible representations of connected compact Lie groups whose orbit space is isometric to that of a representation of a finite extension of $Sp(1)^k$ for some $k>0$. It follows that they are obtained from isotropy representations of certain quaternion-K\"ahler symmetric spaces by restricting to the "non-$Sp(1)$-factor".