Naive Lie Theory
《Naive Lie Theory》 · 印刷第 256—257 页 Section 15.3「The root datum of an F-reductive group」，15.3.1 的群、环面及特征群约定与相对根系命名句
In this section $G$ is a connected $F$-reductive group. We fix a maximal $F$-split $F$-subtorus $S$ of $G$. […] We denote by ${}_F X$ and ${}_F X^\vee$ the group of characters, respectively cocharacters, of $S$ and by $\langle\, ,\,\rangle$ the pairing between them. By 13.2.2 (i) the characters and cocharacters of $S$ are defined over $F$. We assume that $S$ is non-trivial, so ${}_F X\ne\{0\}$. If $H$ is a subgroup of $G$ normalized by $S$, we denote by $R(H,S)$ the set of non-zero weights of $S$ in $L(H)$. The characters in $R(G,S)$ are the roots of $G$ relative to $S$, or the $F$-roots of $G$. They form a subset ${}_F R$ or ${}_F R(G)$, ${}_F R(G,S)$ of ${}_F X$. The Weyl group of $G$ relative to $F$ is the finite group ${}_F W=N_G(S)/Z_G(S)$ (see 3.2.9). We view it as a group of automorphisms of ${}_F X$. It will follow from 15.3.8 that ${}_F R$ (if non-empty) is a root system (the relative root system) with Weyl group ${}_F W$ (the relative Weyl group).