Past exam of the representation theory of semisimple Lie algebras course of the University of Oxford / 2010 / 1 / b

For the intended module, index a basis by $v_k=a_{l-2k}$ for $0\le k\le l$, and usewith $v_{-1}=v_{l+1}=0$. This makes the basis endpoints distinct, identifies the lowering operator, and includes its missing raising action at $k=l$. These are the necessary indexing corrections to the printed display. The same definition for $l=0$ supplies the trivial module needed for classification.

Directly on $v_k$,These hold at the endpoints too, proving the three defining ${\mathfrak{sl}}_2$ relations. FinallyThus the Casimir element acts by that scalar on the whole module.