On the Meromorphic Extension of the Spherical Functions on Noncompactly Causal Symmetric Spaces

We determine integral formulas for the meromorphic extension in the λ-parameter of the spherical functions ϕλ on a noncompactly causal symmetric space. The main tool is Bernsteinʹs theorem on the meromorphic extension of complex powers of polynomials. The regularity properties of ϕλ are deduced. In particular, the possible λ-poles of ϕλ 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 cΩ-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 ϕλ. 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 λ-singularities are located in this general context.