Let G be a connected, real, semisimple Lie group contained in its complexification GC , and let K be a maximal compact subgroup of G. We construct a KC -G double coset domain in GC , and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. .
