Can anyone help me understand why there are no nontrivial homomorphisms $\text{Sp}(n,1) \to \text{GL}(d,\mathbb{R})$ if $n$ is sufficiently large relative to $d$?
By definition, $\text{Sp}(n,1)$ is the isometry group of $n$-dimensional quaternionic hyperbolic space. (By the way, does anyone know of a definition in terms of matrices or more familiar Lie groups?)