An exotic group $C^*$-algebra of a non-amenable group $G$ is a $C^*$-completion of $C_c(G)$ that lies naturally in between the reduced and the universal group $C^*$-algebra of $G$. Using asymptotic properties of matrix coefficients of representations of $G$, one can construct potentially exotic group $C^*$-algebras. For simple Lie groups with real rank one, this construction was shown to define exotic group $C^*$-algebras, by Samei and Wiersma and by de Laat and Siebenand. I will explain how similar methods can be used to show the existence of exotic group $C^*$-algebras of higher rank simple Lie groups, e.g. $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$.