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 Makareviˇc spaces of type I, and the satellite cones. © 2004 Elsevier Inc. All rights reserved.