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 ×^m . As a consequence, we can also compute the heat kernel associated with the weighted ∂̄ -equation in ℂⁿ when the weight is given by exp(-φ(z,z) λ) where φ:ℂⁿ ×ℂ ⁿ →ℂ^m is a quadratic, sesquilinear form and λ ∈ ℝ^m . Our method involves the representation theory of the Lie group M and the group Fourier transform.