[@MR757954], but work over ${\overline{{\mathbb{Q}}}}$. See also section 3.3 of Exposé XXIV of [@MR0207710], which is perhaps where the trick of taking limits of based root data is first introduced.

Let $k$ be a field and let $G$ be a connected reductive algebraic group over $k$. Fix once and for all a separable closure $k^{{\operatorname{sep}}}$ of $k$, and let $\Gamma_k$ denote ${\operatorname{Gal}}(k^{{\operatorname{sep}}}/k)$. The group $G$ splits over $k^{{\operatorname{sep}}}$, and if we choose a maximal torus $T$ in $G_{k^{{\operatorname{sep}}}}$ and a Borel subgroup $B$ of $G_{k^{{\operatorname{sep}}}}$ containing $T$, one can associate the based root datum $\Psi_0(G,B,T):=(X^*(T),\Delta^*(B),X_*(T),\Delta_*(B))$ consisting of the character and cocharacter groups of $T$, and the roots and coroots which are simple and positive with respect to the ordering defined by $B$. Now let $Z_G$ denote the centre of $G$; then $(G/Z_G)(k^{{\operatorname{sep}}})$ acts on $G_{k^{{\operatorname{sep}}}}$ by conjugation, and if $T'$ and $B'$ are another choice of maximal torus and Borel then there is an element of $(G/Z_G)(k^{{\operatorname{sep}}})$ sending $B'$ to $B$ and $T'$ to $T$, and all such elements induce the same isomorphisms of based root data $\Psi_0(G,B,T)\to\Psi_0(G,B',T')$. Following Kottwitz, we define $\Psi_0(G):=(X^*,\Delta^*,X_*,\Delta_*)$ to be the projective limit of the $\Psi_0(G,B,T)$ via these isomorphisms. This means in practice that given 