  We study spectral properties of sub-Riemannian Laplacians, which are
hypoelliptic operators. The main objective is to obtain quantum ergodicity
results, what we have achieved in the 3D contact case. In the general case we
study the small-time asymptotics of sub-Riemannian heat kernels. We prove that
they are given by the nilpotentized heat kernel. In the equiregular case, we
infer the local and microlocal Weyl law, putting in light the Weyl measure in
sR geometry. This measure coincides with the Popp measure in low dimension but
differs from it in general. We prove that spectral concentration occurs on the
shief generated by Lie brackets of length r-1, where r is the degree of
nonholonomy. In the singular case, like Martinet or Grushin, the situation is
more involved but we obtain small-time asymptotic expansions of the heat kernel
and the Weyl law in some cases. Finally, we give the Weyl law in the general
singular case, under the assumption that the singular set is stratifiable.
