In this talk (based on a joint work with Bezrukavnikov) I will discuss the representation theory of semisimple Lie algebras $\mathfrak g$ in very large positive characteristic $p$. It is classically known that the universal enveloping algebra is a free ﬁnite rank module over its central subalgebra known as the $p$-center. This subalgebra is a copy of the symmetric algebra of the initial Lie algebra (with Frobenius twist). In particular, all irreducible representations are ﬁnite dimensional and have a $p$-character, an element of $\mathfrak g$. A particularly interesting case is when the $p$-character is nilpotent. It is known after Bezrukavnikov and collaborators that the set of simples with nilpotent $p$-character is independent of $p$ and the dimensions are polynomials in $p$ as long as $p$ is sufficiently large. In this talk I will explain how to compute the degrees of these dimension polynomials.