Root Datum of a Reductive Group

Let \(G\) be a connected reductive group over an algebraically closed field, and let \(T\subseteq G\) be a maximal torus. Write \[ X=X^*(T),\qquad X^\vee=X_*(T). \] The adjoint action of \(T\) on \(\operatorname{Lie}(G)\) gives a finite set of nonzero weights \(\Phi\subset X\), called the roots. For each \(\alpha\in\Phi\), the corresponding rank-one subgroup determines a coroot \(\alpha^\vee\in X^\vee\). The quadruple \[ (X,\Phi,X^\vee,\Phi^\vee) \] is a root datum.

Changing the maximal torus \(T\) conjugates the construction, so the isomorphism class of the root datum depends only on \(G\).

The Weyl group \[ W(G,T)=N_G(T)/C_G(T) \] acts faithfully on \(X\) and is generated by the reflections \[ s_\alpha(x)=x-\langle x,\alpha^\vee\rangle\alpha. \]