A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group

Let $\mathcal{L}_\gamma = -\frac{1}{4} \left( \sum_{j=1}^n(X_j^2+Y_j^2)+i\gamma T \right)$ where $\gamma \in \mathbb{C}$, and $X_j$, $Y_j$ and $T$ are the left-invariant vector fields of the Heisenberg group structure for $\mathbb{R}^n \times \mathbb{R}^n\times \mathbb{R}$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the heat equation $\partial_s\rho = -\mathcal{L}_\gamma\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted $\bar{\partial}$-operator in $\mathbb{C}^n$ with weight $\exp(-\tau P(z_1,\dots,z_n))$, where $P(z_1,\dots,z_n) = \frac 12(\vert\operatorname{Im}z_1\vert^2 + \cdots +\vert\operatorname{Im} z_n\vert^2)$ and $\tau\in\mathbb{R}$.