Let G be a real connected
noncompact semisimple Lie group with finite center; we shall denote the algebras of
Lie groups L by the corresponding lower case German letter, l. We assume that if Gc
is the simply connected complex analytic Lie group with Lie algebra gc (here for
any vector space V defined over R we denote its complexification by Vc; in
particular gc is the complexification of g) then G ⊂ Gc. Fix a maximal compact
subgroup K of G. Assume further that rk(G∕K) = 1. This paper has two
principal sections. In §I we characterize the invariant transforms of functions in
𝒞p(G : F)(F ⊂ K,|F| < ∞); §II deals with the characterization of the orbital
integrals of such functions.
