Holomorphic extensions of representations:: (I) automorphic functions

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-infinity 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 Maabeta forms.