  We present an invariant definition of the hypoelliptic Laplacian on
sub-Riemannian structures with constant growth vector, using the Popp's volume
form introduced by Montgomery. This definition generalizes the one of the
Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant
problems on unimodular Lie groups we prove that it coincides with the usual sum
of squares.
  We then extend a method (first used by Hulanicki on the Heisenberg group) to
compute explicitly the kernel of the hypoelliptic heat equation on any
unimodular Lie group of type I. The main tool is the noncommutative Fourier
transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the
metrics inherited by the Killing form), and the group SE(2) of rototranslations
of the plane.
  Our study is motivated by some recent results about the cut and conjugate
loci on these sub-Riemannian manifolds. The perspective is to understand how
singularities of the sub-Riemannian distance reflect on the kernel of the
corresponding hypoelliptic heat equation.
