  We derive an integral representation for the subelliptic heat kernel of the
complex anti-de Sitter fibration. Our proof is different from the one used in
Jing Wang \cite{Wan} since it appeals to the commutativity of the D'Alembertian
and of the Laplacian acting on the vertical variable rather than the analytic
continuation of the heat semigroup of the real hyperbolic space. Our approach
also sheds the light on the connection between the sub-Laplacian of the above
fibration and the so-called generalized Maass Laplacian, and on the role played
by the odd dimensional real hyperbolic space.
