Invariant trilinear forms on spherical principal series of real rank one semisimple lie groups

Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form K ν indexed by a complex multiparameter ν = (ν1, ν2, ν3) and defined on the space of smooth functions on the product of three spheres in Fⁿ, where F is either ℝ ℂ ℍ or O with n = 2. We then study the analytic continuation of the trilinear form with respect to (ν₁, ν₂, ν₃), where we locate the hyperplanes containing the poles. Using a result due to Johnson and Wallach on the so-called «partial intertwining operator», we obtain an expression for the generalized Bernstein-Reznikov integral K ν (1⊗1⊗1) in terms of hypergeometric functions.