Character and cocharacter lattices

Let $T$ be a torus, i.e. a compact connected abelian Lie group. Write $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ for the circle group.

• The character lattice of $T$ is $X^{\ast }(T):=\mathrm{Hom}(T,S^1),$ the group of continuous (equivalently smooth) group homomorphisms $T\to S^1$, with pointwise multiplication.
• The cocharacter lattice of $T$ is $X_{\ast }(T):=\mathrm{Hom}(S^1,T),$ the group of continuous group homomorphisms $S^1\to T$, with pointwise multiplication.
• Pairing. For $\chi \in X^{\ast }(T)$ and $\eta \in X_{\ast }(T)$ the composite $\chi \circ \eta :S^1\to S^1$ is a continuous homomorphism; every such homomorphism has the form $z\mapsto z^n$ for a unique $n\in \mathbb{Z}$, and the pairing is $⟨\chi ,\eta ⟩:=n,\text{where}\chi (\eta (z))=z^n\mathrm{.}$ The pairing is biadditive in $\chi$ and $\eta$.

The identity element of either lattice is the trivial homomorphism; the inverse of $\chi$ is the character $t\mapsto \chi (t{)}^{-1}$ for $t\in T$ and is written ${\chi }^{-1}$, while the inverse of $\eta$ is the cocharacter $z\mapsto \eta (z{)}^{-1}$ and is written ${\eta }^{-1}$. Both lattices are abelian groups.

• Both lattices are finitely generated free abelian groups: for $T\cong (S^1{)}^r$ one has $X^{\ast }(T)\cong {\mathbb{Z}}^r$ and $X_{\ast }(T)\cong {\mathbb{Z}}^r$, and the pairing becomes the standard dot product in the dual coordinates. This is proved on this page by differentiating characters; the definition itself asserts no freeness.
• The duality between $X^{\ast }$ and $X_{\ast }$ is perfect: a character is trivial exactly when it pairs to zero with every cocharacter and a cocharacter is trivial exactly when it pairs to zero with every character. This is proved with the differentiation theorem on this page.
• In the following, characters of a maximal torus $T$ are written multiplicatively, and the integer $n$ of the pairing is the weight of the cocharacter; the roots of $(G,T)$ are characters, so they pair integrally with the cocharacters supplied by the compact root $SU(2)$ subgroups.