  The deep theory of approximate subgroups establishes 3-step product growth
for subsets of finite simple groups $G$ of Lie type of bounded rank. In this
paper we obtain 2-step growth results for representations of such groups $G$
(including those of unbounded rank), where products of subsets are replaced by
tensor products of representations.
  Let $G$ be a finite simple group of Lie type and $\chi$ a character of $G$.
Let $|\chi|$ denote the sum of the squares of the degrees of all (distinct)
irreducible characters of $G$ which are constituents of $\chi$. We show that
for all $\delta>0$ there exists $\epsilon>0$, independent of $G$, such that if
$\chi$ is an irreducible character of $G$ satisfying $|\chi| \le
|G|^{1-\delta}$, then $|\chi^2| \ge |\chi|^{1+\epsilon}$. We also obtain
results for reducible characters, and establish faster growth in the case where
$|\chi| \le |G|^{\delta}$.
  In another direction, we explore covering phenomena, namely situations where
every irreducible character of $G$ occurs as a constituent of certain products
of characters. For example, we prove that if $|\chi_1| \cdots |\chi_m|$ is a
high enough power of $|G|$, then every irreducible character of $G$ appears in
$\chi_1\cdots\chi_m$. Finally, we obtain growth results for compact semisimple
Lie groups.
