Highest-weight theorem

Fix a complex semisimple Lie algebra $\mathfrak g$, a Cartan subalgebra $\mathfrak h$, and a choice of positive roots (hence simple roots and fundamental weights).

Theorem (Highest-weight classification, Lie algebra form).

1. Every finite-dimensional irreducible $\mathfrak g$-module is a highest-weight representation with a unique highest weight $\lambda\in\mathfrak h^*$.
2. A weight $\lambda$ occurs as the highest weight of a finite-dimensional irreducible module if and only if $\lambda$ is dominant integral (i.e. it pairs with all simple coroots to give nonnegative integers).
3. For each dominant integral $\lambda$, there exists (up to isomorphism) a unique finite-dimensional irreducible $\mathfrak g$-module $V(\lambda)$ with highest weight $\lambda$.

Group form (compact groups).
If $G$ is a compact connected Lie group with maximal torus $T$ (see maximal tori ), then irreducible unitary representations of $G$ are similarly classified by dominant integral weights of $T$; differentiating recovers the Lie-algebra classification (compare differentiation of representations ).

Context.
This theorem is the conceptual reason that objects like fundamental representations (highest weights equal to fundamental weights) play a foundational role: they correspond to the vertices of the Dynkin diagram and generate the dominant cone.