Decomposition of the space of polynomials

Question

I want to show that I can decompose the vector space of polynomials with $n$ variables over $\mathbb{C}$ in the following methods. For a harmonic and homogeneous polynomial $f$, let $V(f)$ be spanned by $\{f,yf,y^2f,...\}$, where $y=x_1^2+\cdots+x_n^2$. Then let $f_1,\cdots,f_m$ be a basis of the space of harmonic and homogeneous polynomials of degree $k$ and let $V(k)=V(f_1)+\cdots+V(f_m)$. I want to show that $V(k)=V(f_1)\oplus\cdots\oplus V(f_m)$ and $\mathbb{C}[x_1,\cdots,x_n]=\bigoplus\limits_{i=0}^\infty V(i)$.

Answer 1

Just a sketch. Note that the operators $x=\Delta$(Laplacian), $y=-\frac{1}{4}(x_1^2+\cdots+x_n^2)$ and $h=-(x_1\frac{\partial}{\partial x_1}+\cdots+x_n\frac{\partial}{\partial x_n}+\frac{n}{2})$ and their actions on $\mathbb{C}[x_1,\cdots,x_n]$ form a representation of Lie algebra $sl_2(\mathbb{C})$. Moreover, $\{y^kh_0|k\in\mathbb{N}\}$ is an irreducible representation for $h_0$ harmonic and homogeneous.

Then to write a polynomial in the sum of above form, it is sufficient to show that a polynomial $f$ can be written as $f_0+yg$, where $f_0$ is harmonic and $g\in\mathbb{C}[x_1,\cdots,x_n]$. To show this, it is sufficient to deal with the homogeneous situation. Note that the degree of $f$ is finite. Hence there is some integer $i$ s.t. $x^if=0$. Using induction on $i$, show that there is a constant $c$ s.t. $x^{i-1}(f-cyxf)=0$.

After that, it is easier to show that the sum is a direct sum with the help of the structure of Lie algebra.