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

Markov semi-groups associated with the complex unimodular group $Sl(2,\mathbb{C})$