  Boundary conditions for Bismut's hypoelliptic Laplacian which naturally
correspond to Dirichlet and Neumann boundary conditions for Hodge Laplacians
are considered. Those are related with specific boundary conditions for the
differential and its various adjoints. Once the closed realizations of those
operators are well understood, the commutation of the differential with the
resolvent of the hypoelliptic Laplacian is checked with other properties like
the PT-symmetry, which are important for the spectral analysis.
