Let $F\twoheadrightarrow G$ be an epimorphism of groups, $F$ being finitely generated and free. Let $H$ be its kernel. Consider a lifting $i:G\hookrightarrow F$ of the epimorphism.
Every element of $C[F]$ is of the form
$a=\sum_{g\in G} a(g)\,i(g)$, where $a(g)\in C[H]$.
For any character $\chi$ of $H$ it is easy to see that
$$\|a\|_{\max}^2\;\ge\;\max\left\{\sum_{g\in G}\left|\,\chi(a(g))\right|^2\, ,\,\sum_{g\in G} \left|\,\chi\!\left(g^{-1}(a(g))\right)\right|^2\right\}$$
(here the action of $G$ is induced by the adjoint action of $F$).
Proof: Consider the action of $a$ and $a^*$ in the representation induced by the character.
Question: Can this estimate be improved?
For example, could the following be true: Fix an integer $p$. Let $w(g)$ be the partial word of length $p$ in $i(g)$. Then
$$\|a\|^2_{\max}\ge C_p\sum\left|\,\chi\left(w(g)^{-1}(a(g))\right)\right|^2$$
with $C_p$ polynomial in $p\,$?
What should I read in order even to approach proving/disproving such things?