# Expression of the Laplacian of the reduced Heisenberg group?

Let $\mathbb C^n$ be the n-dimensional complex field endowed with a positive definite hermitian form $H(z,w)$. The corresponding symplectic form is $E(z,w)= \Im (H(z,w))$, where $\Im$ denotes the imaginary part of complex numbers.

If we consider the set $H=\mathbb C^n \times \mathbb{U}(1)$, and define on it a law of combination \begin{align} (z,\lambda).(w,\mu) &= \left(z+w, \lambda \mu \, e^{i E(z,w)}\right) \end{align} then $H$ becomes a two step nilpotent group we will call the reduced Heisenberg group.

I will want to know what the expression of the Laplacian $\Delta$ on $H$ ?

• No, this is not the case, I seek the expression of $\Delta$ of $H=\mathbb C^n \times \mathbb{U}(1)$, It is not of $H=\mathbb C^n \times \mathbb R$. – Z. Alfata May 10 '16 at 15:56