I am stuck on an exercise.
(a) Let $\mathfrak{X}$ be an irreducible $F$-representation of $G$ over an arbitrary field. Show that $\sum_{g \in G} \mathfrak{X}(g) = 0$ unless $\mathfrak{X}$ is the principal representation.
(b) Let $H \subset G$ and $g \in G$ be such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb{C}$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show that $\chi(g) = 0$.
For the second part, the hint given is to compute the trace of $\sum_{h \in H} \mathfrak{X}(hg)$, where $\mathfrak{X}$ affords $\chi$. I see that this is equal $|H| \chi(g)$, so it suffices to show that it vanishes, but I have not been able to do this.
Any hints or help would be appreciated, thanks.