Weight space

For a representation relative to a Cartan subalgebra, is the simultaneous eigenspace with weight .

Weight space

Definition

Let $\mathfrak g$ be a complex Lie algebra, $\mathfrak h\subseteq\mathfrak g$ an abelian subalgebra (typically a Cartan subalgebra in the semisimple setting), and let $\rho:\mathfrak g\to\mathfrak{gl}(V)$ be a representation. For $\lambda\in\mathfrak h^\ast$ , the $\lambda$ -weight space is

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

If $V_\lambda\neq 0$ , then $\lambda$ is a weight of the representation .

Interaction with roots (semisimple context)

When $\mathfrak g$ is semisimple and $\mathfrak h$ is a Cartan subalgebra, $\mathfrak g$ decomposes into root spaces $\mathfrak g_\alpha$ (see root space decomposition ). For $X\in \mathfrak g_\alpha$ and $v\in V_\lambda$ , one has

X\cdot v \in V_{\lambda+\alpha},

so root vectors “shift” weights by roots.

Context

For finite-dimensional representations of semisimple Lie algebras, the action of $\mathfrak h$ is semisimple and one gets a direct sum decomposition

V=\bigoplus_\lambda V_\lambda.

This weight-space decomposition is one of the main inputs to highest-weight methods and depends crucially on complete reducibility phenomena (compare Weyl’s complete reducibility theorem ).