Let $\mathbb{H}$ denote the quaternions. Let $G$ be a group, and define a *representation of $G$ on $\mathbb{H}^n$* in the natural way; that is, its a map $\rho:G\rightarrow Hom_{\mathbb{H}-}(\mathbb{H}^n,\mathbb{H}^n)$ such that $\rho(gg')=\rho(g)\rho(g')$ (where $Hom_{\mathbb{H}-}$ denotes maps as left $\mathbb{H}$-modules). Representations of algebras and Lie algebras can be defined in a similar way.

Any quaternionic representation of $G$/$R$/$\mathfrak{g}$ can restrict to a complex representation by choosing a $\gamma\in \mathbb{P}^1$ such that $\gamma^2=-1$, and using $\gamma$ to define a map $\mathbb{C}\hookrightarrow \mathbb{H}$. In this way, any quaternionic representation gives a $\mathbb{C}\mathbb{P}^1$-family of complex representations which parametrize the choice of $\gamma$. Furthermore, since every element in $\mathbb{H}$ is in the image of some inclusion $\mathbb{C}\hookrightarrow \mathbb{H}$, this family of complex representations determines the quaternionic representation.

This observation almost seems to imply that there is nothing interesting to say about quaternionic representations that wouldn't come up while studying complex representations. However, this is neglecting the fact that there might be interesting information in *how* the $\mathbb{C}\mathbb{P}^1$-family of complex representations is put together.

For instance, any finite group will have a discrete set of isomorphism classes of complex representations, and so any quaternionic representation will have all complex restrictions isomorphic. However, the quaternion group $\mathbf{Q}:= ( \pm1,\pm i, \pm j,\pm k)$ has an 'interesting' quaternionic representation on $\mathbb{H}$ (more naturally, it is a representation of the opposite group $\mathbf{Q}^{op}$ by right multiplication, but $\mathbf{Q}^{op}\simeq \mathbf{Q}$).

My question broadly is: **What other groups, rings and Lie algebras have quaternionic representations that are interesting (in some non-specific sense)?**

This question came up when I was reading a paper of Kronheimer's, where he describes a non-canonical hyper-Kahler structure on a coadjoint orbit of a complex semisimple Lie algebra. At any point in such a coadjoint orbit determines a representation of $\mathfrak{g}$ on the tangent space to the coadjoint orbit, which by the hyper-Kahler structure is naturally a quaternionic vector space. I wondered if this representation could be 'interesting', and then realized I had no real sense of what an interesting quaternionic representation would be.