The SO(n) invariant subspaces of polynomials of the creation operators are completely characterized by the theory of spherical harmonics, see for example, the following exposition.

Summary of the construction, Let:

$ d(z) = z_1^2+ . . . + z_n^2$

$ \Delta = (\frac{\partial}{\partial z_1})^2 + . . . + \frac{\partial}{\partial z_n})^2$

Let $\mathcal{P}_m$ be the space of homogeneous polynomials of z of degree m, and

$\mathcal{H}_m = \mathcal{P}_m \cap ker(\Delta)$

Then the SO(n) invariant subspaces are given by:

$\mathcal{P}^l_m = d^l \mathcal{H}_m $

The SO(n) invariant subspaces of polynomials of the creation operators are completely characterized by the theory of spherical harmonics, see for example, the following exposition.

Summary of the construction, Let:

$ d(z) = z_1^2+ . . . + z_n^2$

$ \Delta = (\frac{\partial}{\partial z_1})^2 + . . . + \frac{\partial}{\partial z_n})^2$

Let $\mathcal{P}_m$ be the space of homogeneous polynomials of z of degree m, and

$\mathcal{H}_m = \mathcal{P}_m \cap ker(\Delta)$

Then the SO(n) invariant subspaces are given by:

$\mathcal{P}^l_m = d^l \mathcal{H}_m $

Reply to the comment:

$SO(n)$ acts on the linear space spanned by the creation operators according to the fundamental vector representation. The creation operators commute among themselves, thus, one can identify their span (from the point of view of invariant theory) with $C^n$. The components of the vector $z$ in the answer can be identified with the creation perators. The only invariants of the $SO(n)$ action are functions of $d(z)$ (and its complex conjugate). In addition, one can construct invariant subspaces of polynomials of the creation operators, according to the method given above. The action of these polynomial subspaces on the cyclic vector generate the irreducible $SO(n)$ subspaces of eigenfunctions of the isotropic harmonic oscillator Hamiltonian.

Remark:

The homogeneous subspaces $V_{n+2m}$ are not $SO(n)$ irreducible and they decompose according to corollary 1 in the reference.