# Holomorphic extensions of representations: (I) automorphic functions | Annals of Mathematics

Abstract 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. We obtain L∞ 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.

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### 数学年刊（Annals of Mathematics）

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