Representation theory of Lie groups and outer automorphisms

Question

If $G$ is a simply connected Lie group (I have in mind $G=SL_n(\mathbb{C})$), then we have an isomorphism $Aut(G)/Inn(G)\rightarrow Aut(g)/Ad(G)$ induced by taking the differential at $1$; here $g$ is the Lie algebra of $G$, and $Aut(g)$ the group of Lie algebra automorphisms of $g$, and $Ad:G\rightarrow Aut(g)$ the adjoint representation of $g$. Now it is a fact that $SL_2(\mathbb{C})$ has only 1 $2$-dimensional irreducible representation, whereas $SL_3(\mathbb{C})$ has 2 $3$-dimensional representations. Also for $G=SL_2(\mathbb{C})$ we have $\# Aut(G)/Inn(G)=\mathbf{1}$, whereas for $G=SL_3(\mathbb{C})$ we have $\#Aut(G)/Inn(G)=\mathbf{2}$. Surely this cannot be a mere coincidence; what is going on here? Perhaps one can use the number of irreducible representations to deduce the cardinality $\#Aut(G)/Inn(G)$?

Answer 1

If $\varphi : G \to G$ is an automorphism of $G$, then given any representation $\rho : G \to \text{Aut}(V)$, we get another representation $\rho \circ \varphi : G \to \text{Aut}(V)$, which will generally be different if $\varphi$ is an outer automorphism. In particular the two $3$-dimensional representations of $SL_3$ are related by the nontrivial outer automorphism of $SL_3$ in this way. But a priori there could've been other unrelated representations of the same dimension.