Let denote by $H^3$ the 3-dimension Heisenberg group, i.e., the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right),$$ for $z=x+ i y \in \mathbb C$ and $t\in \mathbb R$. I would like to know why $$\chi_{w}(z,t)=e^{i\Re e(w.\bar{z})}; \qquad w\in \mathbb C^{*}, (z,t)\in H^3,$$ is a one-dimensional representation of $H^3$ ?
Thank you in advance
This is a composition of three maps: \begin{align*} \mathbb{H}^3 \ni (z,t) & \mapsto z \in \mathbb{C}, \\ \mathbb{C} \ni z & \mapsto \operatorname{Re}(w\bar{z}) \in \mathbb{R}, \\ \mathbb{R} \ni s & \mapsto e^{is} \in \mathbb{C}^*. \end{align*} Each of them is trivially checked to be a group homomorphism.