Exercise 12, § 1

Let $q\in k$, and let W be a vector space with basis $(e_0, e_1, e_2, . . .)$.

a) Show that there exists a unique representation $\rho_q$ of $\mathfrak{s}$ on W such that

2

We have $\rho_q(C) = 4q$, where $C =H^2+ 2H-4X_-X_+$ (cf. Exerc. 1). The representation $\rho_q$ is simple. The elements $x\in \mathfrak{s}$ such that $\rho_q(x)$ admits an eigenvalue are the multiples of $X_+$. The endomorphism $\rho_q(X_+)$ admits 1 as an eigenvalue, with multiplicity 1.

b) Let $\rho$ be a simple representation of $\mathfrak{s}$ such that $\rho (C) = 4q$ and such that $\rho (X_+)$ has 1 as an eigenvalue. Show that $\rho$ is equivalent to $\rho_q$.