Past exam of the Lie algebras course of the University of Oxford / 2011 / 2 / i / Solution

By the classification of finite-dimensional sl2 representations over $\mathbb{C}$, every finite-dimensional module is a direct sum of simple modules $L_m$, indexed by integers $m\ge 0$. The module $L_m$ has dimension $m+1$ and a basis $v_0,\dots ,v_m$ withwith out-of-range vectors interpreted as zero. In particular $v_0$ is a highest weight vector of weight $m$. The weights are $m,m-2,\dots ,-m$, each with multiplicity one. The multiplicities of the $L_m$ in a finite module determine that module up to isomorphism.