Holomorphic extensions of representations. I: Automorphic functions
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Publication:1883982
DOI10.4007/ANNALS.2004.159.641zbMATH Open1053.22009arXivmath/0210111OpenAlexW2169773130MaRDI QIDQ1883982
Author name not available (Why is that?)
Publication date: 21 October 2004
Published in: (Search for Journal in Brave)
Abstract: 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.
Full work available at URL: https://arxiv.org/abs/math/0210111
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