Theorem of the highest weight

Remarks

For part 1, a weight vector of maximal weight is killed by all positive root vectors, and it generates the irreducible module. Integrality and dominance follow from the representation theory of the copy of $\mathfrak{sl}_2$ attached to each simple root. Part 2 holds because an irreducible module generated by a highest weight vector of weight $\lambda$ is a quotient of the Verma module $M(\lambda)$, which has a unique irreducible quotient. For part 3, one shows that this quotient $L(\lambda)$ is finite-dimensional when $\lambda$ is dominant integral, since its set of weights is then stable under the Weyl group. Example: for $\mathfrak{sl}_2(\C)$ the dominant integral weights are $\lambda(h) = n \in \Z_{\ge 0}$, and $L(n)$ is the irreducible module of dimension $n + 1$. Dominance is necessary: for $\lambda(h) = -1$, the irreducible quotient of $M(\lambda)$ is $M(\lambda)$ itself, which is infinite-dimensional.

Going deeper

For a student

With Weyl's theorem on complete reducibility, the theorem describes every finite-dimensional module as a sum of modules $L(\lambda)$. The Weyl character formula then gives the weights and multiplicities of $L(\lambda)$. A similar classification holds for compact connected Lie groups, with highest weights that are integral for the group. For a simply connected group, all dominant integral weights occur. In the language of groups, the Borel–Weil theorem realises $L(\lambda)$ as sections of a line bundle on the flag variety.