Let \(G\) be a group acting on a variety \(X\) over \(\mathbb C\). An eigenfunction of \(X\) is a function \(f\in \mathcal O_{X}\) such that for each \(g\in G\)
\begin{align*} (g\cdot f)(x) = \chi(g)\cdot f(x) \end{align*}for some character \(\chi\in \Hom_{\text{alg-grp}}(G,\mathbb C^{*})\).