Tensor Power Asymptotics for Linearly Reductive Groups

Given a finite-dimensional faithful representation $V$ of a linearly reductive group $G$ over a field $K=\bar{K}$, we consider the growth of the number of irreducible factors of $V^{\otimes n}$ when $n$ is large. We prove that there exist upper and lower bounds which are constant multiples of $n^{-u\mathrm{/}2}(\dim V{)}^n$, where $u$ is the dimension of any maximal unipotent subgroup of $G$.