Exercise 10, § 1

Let $m$ be an integer $>0$, $\mathbf{S}$ the graded algebra $k[X_1,\ldots,X_m]$, and $a_1,\ldots,a_m$ elements of $k^*$. Let $\Phi=\displaystyle\sum_{i,j=1}^{m}a_{ij}X_iX_j$ be a quadratic form; put

where $(b_{ij})$ is the inverse matrix of $(a_{ij})$. a) Show that there exists a unique $\mathfrak{s}$-module structure on $\mathbf{S}$ such that $X_+f=$ $\frac{1}{2}D(f), X_-f=\frac{1}{2}\Phi f$ and $Hf=-(m/2 +n)f$ if $f$ is homogeneous of degree $n$. b) Put $A^n=\mathbf{S}^n\cap$ Ker(D). Show that $\mathbf{S}^n$ is the direct sum of the subspaces $\Phi^p.A^q$ for $2p+q=n$. Deduce the identity

$\sum_{n=0}^{\infty}$ dim(A$^n$)$T^n= (1 + T)/(1-T)^{m-1}$.

c) If $f$ is a non-zero element of $A^n$, the $\Phi^pf,p\geq 0$, form a basis of a simple $\mathfrak{s}$-submodule of $\mathbf{S}$, isomorphic to the module $Z(-\frac{m}{2}-n)$ of Exerc. 2.

d) Make the cases $m= 1$ and $m= 2$ explicit. Use the case $m= 3$ to recover the results of Exerc. 9.