Weights in the dual Cartan

Let $\mathfrak g$ be a complex semisimple Lie algebra and $\mathfrak h\subset\mathfrak g$ a Cartan subalgebra. A weight of a representation is, by definition, an element of the dual space $\mathfrak h^\ast$ , i.e. a linear functional on $\mathfrak h$ . The root system $\Delta$ of $\mathfrak g$ is also a subset of $\mathfrak h^\ast$ (see root systems), so weights and roots live in the same ambient vector space and can be compared geometrically.

Integral and dominant weights (standard semisimple setup)

Fix a set of simple roots and corresponding coroots $H_\alpha\in\mathfrak h$ (defined so that $\beta(H_\alpha)$ are the Cartan integers). The integral weight lattice is

P=\{\lambda\in\mathfrak h^\ast\mid \lambda(H_\alpha)\in\mathbb Z\ \text{for all simple roots }\alpha\}.

A weight is dominant if $\lambda(H_\alpha)\ge 0$ for all simple roots. Highest-weight classification says finite-dimensional irreducibles are parametrized by dominant integral weights (compare highest-weight classification).

Link with compact groups and maximal tori

If $G$ is a compact connected Lie group with maximal torus $T$ (see the maximal torus theorem), then characters $T\to U(1)$ differentiate to linear functionals on $\mathfrak t=\mathrm{Lie}(T)$ , producing an integral lattice in $\mathfrak t^\ast$ . After complexification, its restriction to the Cartan of the semisimple part matches the integral weights in $\mathfrak h^\ast$ for the complexified semisimple Lie algebra. The Weyl group acts on these lattices and preserves integrality.