More about Wilson’s functional equation

Abstract

Let G be a group with an involution $x \mapsto x^*$, let $\mu :G \rightarrow \mathbb {C}$ be a multiplicative function such that $\mu (xx^*) = 1$ for all $x \in G$, and let the pair $f,g:G \rightarrow \mathbb {C}$ satisfy that

$$\begin{aligned} f(xy) + \mu (y)f(xy^*) = 2f(x)g(y), \ \forall x,y \in G. \end{aligned}$$

For G compact we obtain: If g is abelian, then f is abelian. For G nilpotent we obtain: (1) If G is generated by its squares and $f \ne 0$, then g is abelian. (2) If g is abelian, but not a multiplicative function, then f is abelian.