Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 2 / iii / Solution

A highest-weight vector is a nonzero weight vector $v$ such that ${\mathfrak{g}}_{\alpha }v=0$ for every positive root $\alpha$. Its weight of a representation is a highest weight. For an irreducible finite-dimensional Lie algebra representation, this is the unique highest weight of a representation, and $v$ generates the module; a reducible module can have several highest weights of different summands.

Here is an existence argument for every nonzero finite-dimensional module $V$. For each simple root, its root operators and coroot form an sl2 subalgebra associated with a root. The classification of finite-dimensional sl2 representations shows that this coroot acts diagonalizably with integral eigenvalues. These commuting coroots span the Cartan subalgebra, so simultaneous diagonalization gives a finite weight-space decompositionChoose a real linear functional $\ell$ on the weight lattice span that is positive on every simple root, hence on every positive root, and choose a weight $\lambda$ maximizing $\ell$ among the finitely many weights of $V$. If $v\in V_{\lambda }$ and $X_{\alpha }\in {\mathfrak{g}}_{\alpha }$, thenso $X_{\alpha }v$ belongs to $V_{\lambda +\alpha }$ if nonzero. For positive $\alpha$, that would contradict maximality because $\ell (\lambda +\alpha )>\ell (\lambda )$. Thus every nonzero $v\in V_{\lambda }$ is a highest-weight vector. Every nonzero finite-dimensional module therefore has a highest weight. The zero module has no nonzero vector or weight, so it is the vacuous exception to the literal unrestricted wording.