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

Abstract

We determine integral formulas for the meromorphic extension in the *-parameter of the spherical functions . * 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 . * are deduced. In particular, the possible *-poles of . * 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 0 -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 . * . 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 HarishChandra-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 *-singularities are located in this general context. Academic Press

The $\Theta$-spherical transform and its inversion

“…As in [11] and [18], we introduce the following c-functions. For a multiplicity function m on , a root α ∈ + and λ ∈ ᑾ * C we set Example 2.4.…”

“…In [18] we have developed a theory of spherical functions which generalizes Heckman-Opdam's theory of hypergeometric functions associated with root systems and which also includes as special geometric instance the theory of Faraut-Hilgert-Ólafsson (see also [11]). These generalized functions have been called -spherical functions because, for a fixed geometric multiplicity function on a root system , the different geometric situations (e.g.…”

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

“…A natural problem, crucial to understand the Laplace transform on ordered symmetric spaces, is therefore to locate the -singularities of . A strategy to get an explicit solution to the problem has been proposed in [14]. This approach is based on an integral formula of , valid for in a domain E. This formula reduces the problem of meromorphic extension to the construction of Bernstein identities on a family of polynomials (cf.…”

“…[3]). The authors of [14] have stated a conjecture (cf. Conjecture 1) on the shape of these Bernstein identities.…”

“…We introduce and classify the satellite cones in Section 3 and we recall the properties of the spherical functions in Section 4. In Section 5, we provide a counter example to the conjecture of [14] (Lemma 2), and propose a new conjecture (Conjecture 3). The new conjecture seems weaker than the initial one, but is sufficient to study the singularities of .…”