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

Abstract

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 [Formula: see text] indexed by a complex multiparameter [Formula: see text] 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 [Formula: see text] in terms of hypergeometric functions.