Meromorphic extension of the spherical functions on a class of ordered symmetric spaces

We discuss a conjecture of Ólafsson and Pasquale published in (J. Funct. Anal. 181 (2001) 346). This conjecture gives the Bernstein-Sato polynomial associated with the Poisson kernel of the ordered (or non-compactly causal) symmetric spaces. The Bernstein-Sato polynomials allow to locate the singularities of the spherical functions on the considered spaces. We prove that this conjecture does not hold in general, and propose a slight improvement of it. Finally, we prove that the new conjecture holds for a class of ordered symmetric spaces, called both the Makarevič spaces of type I, and the satellite cones.