In my studies I stumbled upon the following functional type equation. I think there's a good chance that it can be solved easily, but also that my knowledge of functional equations is not sufficient.
Assume $x,y \in \mathbb R^2$ and let $x\wedge y:=x_1y_2-x_2y_1$. I am looking for functions $g\colon \mathbb R^2 \times \mathbb R^2 \to \mathbb C$ satisfying $$g(x+a,y+a) = e^{i\frac{B}{2}((y-x)\wedge a)} e^{\frac{B}{2}(\lvert a \rvert^2 + (y+x)\cdot a)} g(x,y) \quad \text{for all }a\in \mathbb R^2.$$ I already know one solution, namely $g(x,y):= e^{\frac{B}{2}(x\cdot y - i x \wedge y)}$. I am highly curious whether there is another (continuous) solution. Maybe someone is aware of a general approach of solving such an equation?