Past exam of the representation theory of semisimple Lie algebras course of the University of Oxford / 2010 / 2 / b / Solution

Write $h_{\alpha }$ for the simple coroot and choose the corresponding normalized ${\mathfrak{sl}}_2$ triple. The order condition meansThe highest vector has $h_{\alpha }$-representation weight $⟨\chi -\rho ,{\alpha }^{\vee }⟩=k-1$. The raising-lowering identity givesOther simple positive root vectors commute with $E_{-\alpha }$ and also kill this vector. Thus it is a highest vector of representation weightThe universal property of the Verma module gives the required homomorphism. It is injective: under the Poincaré-Birkhoff-Witt theorem identification of each Verma module with the free rank-one $U({\mathfrak{n}}_{-})$ module, it is right multiplication by $E_{-\alpha }^k$. The algebra $U({\mathfrak{n}}_{-})$ has no zero divisors, because its associated graded algebra is the polynomial domain $S({\mathfrak{n}}_{-})$. HenceFor $k=0$ this is the identity map, so the equality case of the order is covered.