An immediate application is the existence of the Langlands complex dual group. If $G$ is a connected reductive group over a field $F$ and $\overline{F}/F$ is an algebraic closure, then $G^\vee$ is the connected reductive group over $\mathbb{C}$ whose root datum is dual to that of $G_{\overline{F}}$, i.e. is obtained by interchanging roots with coroots and simple roots with simple coroots.
In the case when $F$ is a local or global field, Langlands discovery of $G^\vee$ and the related $L$-group ${}^LG$ in 1966 lead him to, and these groups figure prominently in: the local and global Langlands correspondences, the general definition of local and automorphic $L$-functions, and the principle of functoriality.