If $G$ is a finite group, the characters of its irreps satisfy $$ \langle \chi_1,\chi_2\rangle := \frac{1}{|G|}\sum_{g\in G} \chi_1(g)\; \overline{\chi_2(g)} = \delta_{\chi_1,\chi_2}. $$

Alexei Latyntsev asked me the following question.

Instead of a group $G$, let us consider a vertex algebra $V$ (let's say a rational unitary VOA).

Is there an analog of the above orthogonality relations which applies to the characters of irreducible $V$-modules? (Or maybe genus one 1-point functions?)