Exercise 2, § 1

Let $\lambda \in k$, let $Z(\lambda )$ be a vector space having a basis $(e_n)$, with $n= 0,1, . .$., and let $X_+, X_-, H$ be the endomorphisms of $Z(\lambda )$ defined by formulas (2) of Prop. 1.

a) Verify that this gives an $\mathfrak{s}$-module structure on $Z(\lambda )$.

b) Assume that $\lambda$ is not an integer $\geq 0$. Show that the $\mathfrak{s}$-module $Z(\lambda )$ is simple.

c) Assume that $\lambda$ is an integer $\geq 0$. Let $Z'$ be the subspace of $Z(\lambda )$ generated by the $e_n,n > \lambda$. Show that $Z'$ is an $\mathfrak{s}$-submodule of $Z(\lambda )$, isomorphic to $Z(-\lambda -2)$, and that the quotient $Z(\lambda )/Z'$ is isomorphic to the simple $\mathfrak{s}$-module $V(\lambda )$. Show that the only $\mathfrak{s}$-submodules of $Z(\lambda )$ are $0,Z'$ and $Z(\lambda )$.