Split reductive groups

A split reductive group over $k$ is a pair $(G,T)$ consisting of a reductive algebraic group $G$ over $k$ (Radical, unipotent radical, semisimple and reductive algebraic groups) and a maximal torus $T\subseteq G$ that is split, i.e. isomorphic over $k$ to ${\mathbf{G}}_m^r$ for some $r$ (Groups of multiplicative type and tori, Borel subgroups, maximal tori and Borel pairs).

The following supplemental existence and field-invariance facts assume the Axiom of Choice (The Axiom of Choice) through their cited suppliers. Since maximal tori exist, any torus lies in one, and maximality is preserved by field extension (Maximal tori, field extensions, normal subgroups and derived groups), the rank $\dim T$ and the semisimple rank $\mathrm{rk}(G\mathrm{/}R(G))$ are well defined. A homomorphism of split reductive groups $(G,T)\to (G^{\mathrm{\prime }},T^{\mathrm{\prime }})$ is a homomorphism of algebraic groups carrying $T$ into $T^{\mathrm{\prime }}$. Because $G$ is affine of finite type, the adjoint representation $\mathrm{Ad}:G\to {\mathrm{GL}}_{\mathfrak{g}}$ (The adjoint representation of an affine group scheme, The Lie bracket from infinitesimals and the adjoint action) restricts to a rational representation of $T$ on $\mathfrak{g}=\mathrm{Lie}(G)$, to which the eigenspace decomposition of diagonalizable groups applies (Representations of diagonalizable groups split into character eigenspaces). We write $X=X(T)$, $X^{\vee }=X_{\ast }(T)$ for the lattices of Character and cocharacter lattices of a split torus. Every reductive group splits over a finite separable extension of the base field; the split hypothesis is kept throughout.

Under this same assumption, the rank of $(G,T)$ is the common dimension of its maximal tori and the semisimple rank is the rank of the semisimple quotient $G\mathrm{/}R(G)$; both are additive invariants of the pair. The split hypothesis is exactly the requirement that the geometric character lattice $X^{\ast }(T)={\mathrm{Hom}}_{k_s}(T_{k_s},{\mathbf{G}}_m)$ have trivial $\mathrm{Gal}(k_s\mathrm{/}k)$-action; it is always a free $\mathbb{Z}$-module of finite rank, and $X(T)=X^{\ast }(T{)}^{\mathrm{Gal}(k_s\mathrm{/}k)}$. The equivalence with splitting is Multiplicative type groups and Galois character modules, which is why the root datum of $(G,T)$ is defined over $k$ rather than only over a finite extension. A homomorphism of split reductive groups need not carry $T$ isomorphically onto $T^{\mathrm{\prime }}$; the induced map on character lattices is then only a homomorphism. An isogeny carrying $T$ onto $T^{\mathrm{\prime }}$ induces an injective character-lattice map with finite cokernel, by the split character anti-equivalence (Split diagonalizable groups are dual to abelian groups); it need not induce an isomorphism even if the two groups have the same abstract root datum. For example, the isogeny $t\mapsto t^p$ of ${\mathbf{G}}_m$ in characteristic $p$ induces multiplication by $p$ on its character lattice $\mathbb{Z}$.