Coroots and the Weight–Coroot Pairing

For a root $\alpha\in \Phi\subset X^*(T)$, the coroot is a cocharacter $\alpha^\vee\in X_*(T)$.

Equivalently, one chooses an element $H_\alpha$ in the (split) Cartan Lie algebra such that $\alpha(H_\alpha)=2$; then for a weight $\lambda\in X^*(T)$, $\langle \lambda,\alpha^\vee\rangle := \lambda(H_\alpha)\in \mathbb{Z}.$

Key uses:

• Dominance: $\lambda$ is dominant iff $\langle \lambda,\alpha^\vee\rangle\ge 0$ for all simple $\alpha$ (see dominant weights ).
• Duality: swapping roots and coroots produces the dual group .