Root Lattice, Weight Lattice, and Isogeny Forms

For a split reductive $G$ with split maximal torus $T$ (see $X^*(T)$ ), the root system is $\Phi\subset X^*(T)$.

The root lattice is $Q:=\mathbb{Z}\Phi \subset X^*(T)$.

The weight lattice is $P:=\{\lambda\in X^*(T)\otimes\mathbb{Q}:\langle \lambda,\alpha^\vee\rangle\in\mathbb{Z}\ \forall \alpha\in\Phi\}$; for a simply connected semisimple group, $X^*(T)=P$.

For semisimple $G$, central isogeny types correspond to intermediate lattices $Q \subset X^*(T) \subset P$ (or dually for cocharacters).

In the letter: the choice of an intermediate lattice $L$ “between roots and weights” controls the form of $G$ and of its dual group .