Markov Semi-groups Associated with the Complex Unimodular Group Sl(2 , C)

In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011) from the restriction of a particular positive definite function on the complex unimodular group SL(2 , C) to two commutative subalgebras of its universal C^⋆-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. We also supply some arguments supporting 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 , R) and to the Landau operator in the complex plane.