The Product Formula for the Spherical Functions on Symmetric Spaces in the Complex Case

Let G be a semisimple noncompact Lie group with finite center and K a maximal compact subgroup ofG andX = G/K the corresponding Riemannian symmetric space of noncompact type. We have a Cartan decomposition g = k+ p and we choose a maximal abelian subalgebra a of p. In what follows, Σ corresponds to the root system of g and Σ+ to the positive roots. We have the root space decomposition g = g0 + ∑ α∈Σ gα. Let n = ∑ α∈Σ+ gα. Denote the groups corresponding to the Lie algebras a and n by A and N respectively. We have the Cartan decomposition G = KAK and the Iwasawa decomposition G = KAN . Let a+ = {H ∈ A : α(H) > 0 ∀ α ∈ Σ+} and A+ = exp(a+). If λ is a complex-valued functional on a, the corresponding spherical function is