A theorem of Borel asserts that $\mathbf{Sp}(n)\mathbf{Sp}(1)$ and $\mathbf{Sp}(n)\mathbf{U}(1)$ act transitively in the sphere $\mathbb{S}^{4n-1}$. How can we describe these actions? Is there some resource where I can learn more about these groups and their actions in $\mathbb{S}^{4n-1}\subset \mathbb{H}^n$?



On transitive actions on the sphere
The definitions are in Morton Curtis, Matrix Groups, p. 27. The group $Sp(n)$ is the group of quaternionic linear transformations of a quaternionic vector space, which preserve a quaternionic bilinear positive definite form (all made precise by Curtis). The group $Sp(n)Sp(1)$ is the group of real linear transformations of a quaternionic vector space induced by those of $Sp(n)$ and by the group $Sp(1)$ of unit quaternion scalars. The subgroup $Sp(n)U(1)$ uses only the unit complex scalars. These all act by linear transformations, by definition, and preserve the real part of the quaternionic bilinear form, i.e. the real inner product. (Again, Curtis has all of the details.) Therefore they preserve the unit sphere in the quaternionic vector space. Moreover, already $Sp(n)$ acts transitively, which is easy to see by looking at the subgroup stabilizing a particular vector $e\_1$, and then induction.


