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Holomorphic extension of representations: (I) automorphic functions

Bernhard Kroetz·Robert J. Stanton
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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.

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Bernhard Kroetz, Robert J. Stanton. 2006-01-10. Holomorphic extension of representations: (I) automorphic functions. https://arxiv.org/abs/math/0210111

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