Maximal Torus and Weight Lattice

A torus over $k$ is a $k$-group that becomes isomorphic to $(\mathbb{G}_m)^r$ over $\bar k$; it is split if it is already $(\mathbb{G}_m)^r$ over $k$.

A maximal torus $T\subset G$ is a torus not properly contained in any larger torus in $G$.

The (character/weight) lattice of $T$ is $X^*(T):=\mathrm{Hom}_k(T,\mathbb{G}_m)$, an abelian group (for split $T$, $X^*(T)\cong \mathbb{Z}^r$).

Example: For $T$ the diagonal torus in $\mathrm{GL}_n$, $X^*(T)\cong \mathbb{Z}^n$ via $(m_i)\mapsto \left(\mathrm{diag}(t_i)\mapsto \prod_i t_i^{m_i}\right)$.