On the meromorphic extension of the spherical functions on noncompactly causal symmetric spaces

We determine integral formulas for the meromorphic extension in the lambda -parameter of the spherical functions phi (lambda) on a noncompactly causal symmetric space. The main tool is Bernstein's theorem on the meromorphic extension of complex powers of polynomials. The regularity properties of phi (lambda) are deduced. In particular. the possible lambda -poles of phi (lambda) are located among the translates of the zeros of the Bernstein polynomial. The translation parameter depends only on the structure of the symmetric space. The expression of the Bernstein polynomial is conjectured. The relation between the Bernstein polynomial and the product formula of the c(Omega)-function is analyzed. The conjecture is verified in the rank-one case. The explicit formulas obtained in this case yield a detailed description of singularities of phi (lambda). In the general higher rank case. the integral formulas are applied to find asymptotic estimates for the spherical functions. In the Appendix. the spherical functions on noncompactly causal symmetric spaces are regarded as a special instance of Harish-Chandra-type expansions associated with roots systems with arbitrary multiplicities. We study expansions obtained by taking averages over arbitrary parabolic subgroups of the Weyl group of the root system. The possible lambda -singularities are located in this general context. (C) 2001 Academic Press.