Let $\mathcal{A} = \{V : \mathbb{U}_n \rightarrow \mathbb{C}\}$ where $\mathbb{U}_n$ is the group of the complex $n$-th roots of the unity. This group naturally acts on $\mathcal{A}$: for any $a \in \mathbb{U}_n$, and any $V \in \mathcal{A}$
$$
  \begin{equation*}
    (a \star V) : b \mapsto V(a^{-1}b)
  \end{equation*}
$$
I define $Fix(V) = \{a \in \mathbb{U}_n,~ a \star V = V\}$ the symmetry group of $V$.

My question is: can we define an equivariant function $\lambda : \mathcal{A} \rightarrow \mathbb{C}$ which measures the "asymmetry" of $V$, i.e., a function such that for all $a,V$
$$
  \begin{align*}
     \lambda(a \star V) &= a \lambda(V) \\
     \lambda(V) = 0 &\Leftrightarrow Fix(V) = \{1\}
  \end{align*}
$$

Is it related to some known problem ?

Thank you.