The C-function for a Semisimple Symmetric Space 1 Spherical Functions for Riemannian Symmetric Spaces

We discuss the product formula for the c-function for a Riemannian symmetric space and the similar formula for a noncompactly causal symmetric space. We derive two functional equations between the c-functions using the product formulas. Finally some examples where the c-functions turn up in analysis on symmetric spaces are given. Let G be a connected semisimple Lie group with nite center. Let K G be a maximal compact subgroup. Then there exists an involution : G ! G such that G = K. The involution is called a Cartan involution on G. Denote by g the Lie algebra of G. Then g = k s where k = fX 2 g j (X) = Xg =: g , and s = g ? := fX j (X) = ?Xg. Deene an inner product on g by (X; Y) := ?Tr(ad(X)ad((Y)): We notice that ad (X) = ?ad ((X)) for all X 2 g. Fix a maximal abelian subalgebra a of s. Then ad(a) is a commutative family of symmetric operators and hence g = z g (a) M 2 g where g = fX 2 g j 8H 2 a : ad(H)X = (H)Xg, and is the set of 2 a n f0g such that g 6 = f0g. Let X 2 a be such that (X) 6 = 0 for all 2 , and deene + = f 2 j (X) > 0g.