The root datum of a split reductive group

Let $(G,T)$ be a split reductive group over $k$ (Split reductive groups). Its root datum is the quadruple $R(G,T)=(X(T),\mathrm{\Phi }(G,T),X_{\ast }(T),{\mathrm{\Phi }}^{\vee }(G,T)),$ where the character and cocharacter lattices are those of Character and cocharacter lattices of a split torus, the roots are the nonzero adjoint characters, and ${\mathrm{\Phi }}^{\vee }=\{{\alpha }^{\vee }:\alpha \in \mathrm{\Phi }\}$ consists of the associated coroots. Its rank is ${\mathrm{rank}}_{\mathbb{Z}}X(T)=\dim T$. For a split Borel pair $(B,T)$, its base $\mathrm{\Delta }(B)$ consists of roots in ${\mathrm{\Phi }}^{+}(B)$ that are not sums of two positive roots; the pair gives the based root datum $(R(G,T),\mathrm{\Delta }(B))$.

Assume the Axiom of Choice inherited from the root-group, Weyl and centre suppliers for the following structural assertions (The Axiom of Choice). The coroots supplied by Root subgroups of a split reductive group are in bijection with the roots and satisfy $⟨\alpha ,{\alpha }^{\vee }⟩=2$. This quadruple is a reduced root datum in the sense of Abstract root data and their Weyl groups, its Weyl group is canonically $W(G,T)$, and $\mathrm{\Delta }(B)$ is a base whose nonnegative integral combinations recover ${\mathrm{\Phi }}^{+}(B)$ (The Weyl group, Borel subgroups and chambers, Combinatorics of a reduced root datum). Its semisimple rank is $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }=\mathrm{rank}G-\dim Z(G)$, and $G$ is semisimple exactly when $\mathbb{Z}\mathrm{\Phi }$ has finite index in $X(T)$ (Centre, radical and semisimple quotient of a reductive group).

The isomorphism class of the unbased root datum is independent of the split Borel pair, but it is not asserted to have a unique abstract isomorphism: Weyl automorphisms already refute that assertion in type $A_1$. For two fixed split Borel pairs $(B,T),(B^{\mathrm{\prime }},T^{\mathrm{\prime }})$, conjugation by $g\in G(k)$ carrying the first pair to the second induces a canonical comparison of their based root data. If $g^{\mathrm{\prime }}$ is another such element, $g^{-1}{g}^{\mathrm{\prime }}$ lies in $N_G(B)\cap N_G(T)=B\cap N_G(T)=T$, by Borel self-normality and the Weyl/Borel correspondence. Conjugation by $T$ acts trivially on its character and cocharacter lattices and preserves the root-coroot labels, so the two induced comparisons agree. Thus uniqueness applies to the comparison attached to the fixed based pairs, while the unbased datum is defined up to isomorphism class. (The Weyl group, Borel subgroups and chambers, Cartan subgroups: conjugacy, density and normalizers)