On the Divisibility of Degrees of Representations of Lie Algebras

Let $\mathfrak{g}$ be a reductive Lie algebra, and $m$ a positive integer. There is a natural density of irreducible representations of $\mathfrak{g}$, whose degrees are not divisible by $m$. For $\mathfrak{g}={\mathfrak{g}\mathfrak{l}}_n$, this density decays exponentially to $0$ as $n\to \mathrm{\infty }$. Similar results hold for simple Lie algebras and Lie groups, and there are versions for self-dual and orthogonal representations.