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.

中文翻译：

---

有关Wilson函数方程的更多信息

令G为具有对合\（x \ mapsto x ^ * \）的组，令\（\ mu：G \ rightarrow \ mathbb {C} \）为一个乘法函数，使得\（\ mu（xx ^ *）对于所有\（x \ in G \）= 1 \），并让对\（f，g：G \ rightarrow \ mathbb {C} \）满足

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

对于G紧凑型，我们得到：如果g为阿贝尔式，则f为阿贝尔式。对于G幂幂，我们得到：（1）如果G是由其平方和\（f \ ne 0 \）生成的，则g是阿贝尔格。（2）如果g是abelian，但不是乘法函数，则f是abelian。