The ${\mathrm{\square }}_b$-heat equation on quadric manifolds

In this article, we give an explicit calculation of the partial Fourier transform of the ${\mathrm{\square }}_b$-heat equation on quadric submanifolds of $M\subset C^n\times C^m$. As a consequence, we can also compute the heat kernel associated to the weighted dbar-equation in $C^n$ when the weight is given by $\exp (-\varphi (z,z)\cdot \lambda )$ where $\varphi :C^n\times C^n\to C^m$ is a quadratic, sesquilinear form and $\lambda \in R^m$. Our method involves the representation theory of the Lie group $M$ and the group Fourier transform.