Exercise 9, § 1

Consider the operation of $\mathfrak{s}$, via the adjoint representation, on its enveloping algebra $U$ and on its symmetric algebra $\mathbf{S}=\bigoplus S^n$.

a) Determine the weights of the $\mathfrak{s}$-module $\mathbf{S}^n$. Deduce the following isomorphisms of $\mathfrak{s}$-modules:

In particular, the elements of $\mathbf{S}^n$ invariant under $\mathfrak{s}$ form a space of dimension $1$ (resp. $0$) if $n$ is even (resp. odd).

b) Show that the subalgebra of $\mathbf{S}$ consisting of the elements invariant under $\mathfrak{s}$ is the polynomial algebra $k[\Gamma]$, where $\Gamma=H^2-4X_-X_+$. The module $\mathbf{S}^n/\Gamma.\mathbf{S}^{n-2}$ is isomorphic to $V(2n)$.

c) Show that the centre of $U$ is the polynomial algebra $k[C]$, where $C$ is the element defined in Exerc. 1 (use $b$) and the Poincaré-Birkhoff-Witt theorem).