  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 $\mathfrak g=\mathfrak{gl}_n$, this
density decays exponentially to $0$ as $n \to \infty$. Similar results hold for
simple Lie algebras and Lie groups, and there are versions for self-dual and
orthogonal representations.
