Root, weight, and isogeny lattices

Let $G$ be a split semisimple group with maximal torus $T$ and root system $\Phi\subset X^*(T)$ . The root lattice and abstract weight lattice are

Q=\mathbb Z\Phi, \qquad P= \{\lambda\in X^*(T)_\mathbb Q: \langle\lambda,\alpha^\vee\rangle\in\mathbb Z \text{ for all }\alpha\in\Phi\}.

The actual character lattice satisfies

Q\subseteq X^*(T)\subseteq P.

Intermediate lattices classify the central isogeny forms with the given root system. The simply connected form has character lattice $P$ , while the adjoint form has character lattice $Q$ .

Dual statement

On cocharacters one has

Q^\vee\subseteq X_*(T)\subseteq P^\vee,

where $Q^\vee$ here denotes the coroot lattice and $P^\vee$ the coweight lattice—not the integral duals of $Q$ and $P$ with the same symbols. The simply connected form has $X_*(T)=Q^\vee$ , while the adjoint form has the full coweight lattice. The Langlands dual group exchanges the two lattice diagrams.

Type A1

Let $\omega$ be the fundamental weight and $\alpha=2\omega$ . Then

Q=2\mathbb Z\omega \subset P=\mathbb Z\omega, \qquad P/Q\simeq\mathbb Z/2\mathbb Z.

Thus $\operatorname{SL}_2\to\operatorname{PGL}_2$ is the central isogeny with kernel $\mu_2$ , and

\widehat{\operatorname{SL}_2}=\operatorname{PGL}_2(\mathbb C), \qquad \widehat{\operatorname{PGL}_2}=\operatorname{SL}_2(\mathbb C).

Relation to the letter

The letter's intermediate lattice $L$ fixes the isogeny form; its dual lattice fixes the dual group's form. Recording only the abstract root system would lose this information.

References

Robert Steinberg, Lectures on Chevalley Groups, AMS, 2016.
A. Borel, Linear Algebraic Groups, second edition, Springer, 1991.