The Heisenberg group $H^3$ is 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, what is the expression of the coadjoint representation $Ad^*$ of $H^3$ ?
Thank you in advance
The coadjoint action $Ad^*$ is the contragredient of $Ad$ and given on $H_3^*$ by $Ad^* (g)=(Ad \, g^{-1})^*$. This can be computed explicitly, e.g., see section $8.4$ on page $203$ here. The start is to compute $$ (Ad \, g)Y=gYg^{-1}=\begin{pmatrix} 1 & a & c \cr 0 & 1 & b \cr 0 & 0 & 1\end{pmatrix}\begin{pmatrix} 0 & x & z \cr 0 & 0 & y \cr 0 & 0 & 0\end{pmatrix} \begin{pmatrix} 1 & a & c \cr 0 & 1 & b \cr 0 & 0 & 1\end{pmatrix}^{-1} $$ explicitly.