Weight of a representation

Let $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra (see semisimple Lie algebras) and let $\mathfrak h\subset \mathfrak g$ be a Cartan subalgebra. For a representation $\rho:\mathfrak g\to \mathfrak{gl}(V)$ , a linear functional $\lambda\in \mathfrak h^\ast$ is called a weight of $V$ if the corresponding weight space

V_\lambda=\{v\in V\mid \rho(H)v=\lambda(H)v\ \text{for all }H\in\mathfrak h\}

is nonzero.

Context

Weights organize the representation theory of semisimple Lie algebras: the set of weights (with multiplicities $\dim V_\lambda$ ) encodes much of $V$ , and irreducible representations are classified by their highest weight (see the highest-weight theorem). The ambient space where weights live is explained in weights in the dual Cartan.

Equivalent characterizations

Equivalently, $\lambda$ is a weight if there exists $0\neq v\in V$ such that every $H\in\mathfrak h$ acts on $v$ by the scalar $\lambda(H)$ (so $v$ is a simultaneous eigenvector for the commuting family $\rho(\mathfrak h)$ ).