  In this paper, we derive the explicit expressions of two Markov semi-groups
constructed by P. Biane in \cite{Bia1} from the restriction of a particular
positive definite function on the complex unimodular group $Sl(2,\mathbb{C})$
to two commutative subalgebras of its universal $C^{\star}$-algebra. Our
computations use Euclidean Fourier analysis together with the generating
function of Laguerre polynomials with index $-1$, and yield
absolutely-convergent double series representations of the semi-group
densities. In the last part of the paper, we discuss the coincidence, noticed
by Biane as well, occurring between the heat kernel on the Heisenberg group and
the semi-group corresponding to the intersection of the principal and the
complementary series. To this end, we appeal to the metaplectic representation
$Mp(4,\mathbb{R})$ and to the Landau operator in the complex plane.
