TITLE: Proofs of a character identity?
QUESTION [7 upvotes]: Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$.  I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \times \cdots \times C_k} \sum_{V_\lambda} d(V_\lambda)^{1-2g} \chi_\lambda(c_1^{-1}\cdots c_k^{-1}) = |C_1|\cdots|C_k|\sum_{V_\lambda} d(V_\lambda)^{2-2g-k}\chi_\lambda(c_1)\cdots \chi_\lambda(c_k)
$$
where the sums with $V_\lambda$ are over all irreducible characters of $G$ and $d(V_\lambda) = \chi_\lambda(1)$ is the dimension of the representation.  I obtained this from messing around with a theorem due to Frobenius (for $g=0$ and later generalized originally using a quantum topology approach - hence the $g$ for genus).
Is there some nice (ideally purely character-theoretic) proof of this identity?

REPLY [8 votes]: Here is another way to view things: it uses the fact that for each complex irreducible character $\chi$ of $G$, there is an algebra homomorphism $\omega_{\chi} : Z(\mathbb{C}G) \to \mathbb{C}$ defined by $\omega_{\chi}(X) = \frac{\chi(X)}{\chi(1)} .$ This is the underlying root (from a modern viewpoint) of the formula of Frobenius quoted in the question, and the formula beloved of Paul Sally quoted in L. Spice's answer.
For $S$ a subset of $G$ invariant under conjugation, let $S^{+} = \sum_{s \in S} s$ in $\mathbb{C}G.$
Hence for each $\lambda$ we find that $$\frac{\chi_{\lambda}(C_{1}^{+}C_{2}^{+} \ldots C_{k}^{+})}{\chi_{\lambda}(1)} = \prod_{i=1}^{k} |C_{i}|\frac{\chi_{\lambda} (c_{i})}{\chi_{\lambda}(1)}.$$
Expressing the element $C_{1}^{+} \ldots C_{k}^{+}$ as a sum of products of group elements explains the formula of the question.