Disjointness of a simple matrix Lie group and its Lie algebra

Abstract

Let $G$ be a connected closed subgroup of ${\mathrm{GL}}_n(\mathbb{C})$ which is simple as a Lie group and which acts irreducibly on ${\mathbb{C}}^n$. Regarding both $G$ and its Lie algebra $\mathfrak{g}$ as subsets of $M_n(\mathbb{C})$, we have $G\cap \mathfrak{g}\ne \varnothing$ if and only if $G$ is a classical group and ${\mathbb{C}}^n$ is a minuscule representation.

7 pages