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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. . | Annals of Mathematics Holomorphic extensions of representations I automorphic functions By Bernhard Kr otz. and Robert J. Stanton Annals of Mathematics 159 2004 641 724 Holomorphic extensions of representations I automorphic functions By Bernhard Krotz and Robert J. Stanton Abstract Let G be a connected real semisimple Lie group contained in its complex-ification 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. We obtain Ly- 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 Maafi forms. Introduction Complex analysis played an important role in the classical development of the theory of Fourier series. However even for Sl 2 R contained in Sl 2 C complex analysis on Sl 2 C has had little impact on the harmonic analysis of Sl 2 R . As the K-finite matrix coefficients of an irreducible unitary representation of Sl 2 R can be identified with classical special functions such as hypergeometric functions one knows they have holomorphic extensions to some domain. So for any infinite dimensional irreducible unitary representation of Sl 2 R one can expect at most some proper subdomain of Sl 2 C to occur. It is less clear that there is a universal domain in Sl 2 C to which the action of G on K-finite vectors of every irreducible unitary representation has holomorphic extension. One goal of this paper is to construct such a domain for a real connected semisimple Lie group G contained in its complexification GC . It is .
