  Let G be a connected, real, semisimple Lie group contained in its
complexification G_C, and let K be a maximal compact subgroup of G. We
construct a K_C-G double coset domain in G_C, 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. We obtain
L^\infty bounds on holomorphically extended automorphic functions on G/K in
terms of Sobolev norms, and we use these to estimate the Fourier coefficients
of combinations of automorphic functions in a number of cases, e.g. of triple
products of Maass forms.
