I am trying to understand the representations of $SO_3(\mathbb{R})$. Consider the space $P_n$ of homogeneous polynomials of degree $n$ in $(x,y,z)$. I want to understand the characters of $V_n = \ker (\Delta \colon P_n \to P_{n-2})$ where $\Delta$ is the classical Laplacian. The characters of $V_n$ are reduced to understand the characters of $P_n$.
However I do not understand how to compute the character of $V_n$. The elements of $SO_3(\mathbb{R})$ are rotations, call them $r_\theta$ the usual 3D rotation matrix. I shall prove that $$\chi_{V_n}(r_\theta) = e^{in\theta} + e^{i(n-1)\theta} + \cdots + e^{-in\theta}$$ and even that these summands are eigenvalues of $r_\theta$ on $V_n$. I know that a basis for $P_n$ is made of $x^\alpha y^\beta z^\gamma$ where $\alpha + \beta + \gamma = n$. Can I deduce something from the explicit action of $r_\theta$ on such elements? Or is there a more suitable basis for this question?