Exercise 6, § 1

If E is a finite dimensional $\mathfrak{s}$-module, and $n$ an integer, denote by $a_n$ the dimension of the eigenspace of $H_E$ relative to the eigenvalue $n$. Denote by $c_E(T)$ the element of $\mathbf{Z}[T,T^{-1}]$ defined by $c_E(T) =\sum_{n\in\mathbf{Z}}a_nT^n$.

a) Define $L_m$ as in Exerc. 5. Show that

dim $L_m=a_m-a_{m+2}$ for every integer $m\geq 0$.

Deduce that $c_E(T) =c_{E'}(T)$ if and only if E and $E'$ are isomorphic. Recover this result by using Exerc. $18e)$ of Chap. VII, §3.

b) Show that $c_{E\oplus F}=c_E+c_F$ and $c_{E\otimes F}=c_E.c_F$.

c) We have $c_{V(m)}(T) = (T^{m+1}-T^{-m-1})/(T-T^{-1})$. d) Deduce from a), b), c) that, if $m\geq m'\geq 0$, the $\mathfrak{s}$-module $V(m)\otimes V(m')$ is isomorphic to

If $m\geq 0$, the $\mathfrak{s}$-module $S^2V(m)$ is isomorphic to

and the $\mathfrak{s}$-module $\Lambda^2V(m)$ is isomorphic to