The □b-Heat Equation on Quadric Manifolds

In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □b-heat equation on quadric submanifolds M⊂ℂn×ℂm. As a consequence, we can also compute the heat kernel associated with the weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline{\partial}$\end{document}-equation in ℂn when the weight is given by exp (−φ(z,z)⋅λ) where φ:ℂn×ℂn→ℂm is a quadratic, sesquilinear form and λ∈ℝm. Our method involves the representation theory of the Lie group M and the group Fourier transform.