Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 102 / 2 / b / Solution

Fix a simple root ${\alpha }_i$ and choose generators $e_i,f_i,h_i$ of the sl2 subalgebra associated with a root, with $h_i={\alpha }_i^{\vee }$ andThe highest-weight vector satisfies $e_{i}v=0$ and $h_{i}v=m_{i}v$, where $m_i=⟨\lambda ,{\alpha }_i^{\vee }⟩$. The classification of finite-dimensional sl2 representations already implies that $m_i$ is a nonnegative integer. One can also see the necessary integrality directly from the lowering operator.

Indeed, $f_i^{k}v$, if nonzero, has $h_i$ eigenvalue $m_i-2k$. These distinct eigenvalues cannot occur indefinitely in a finite-dimensional space, so there is a largest $r\ge 0$ with $f_i^{r}v\ne 0$. The sl2 Lie algebra relations imply, by induction on $k$,For $k=r+1$ the left side is zero, hence $(r+1)(m_i-r)f_i^{r}v=0$. Thus $m_i=r\in {\mathbb{Z}}_{\ge 0}$. Since this holds for every simple root,