Holomorphic extension of representations: (I) automorphic functions

Details Holomorphic extension of representations: (I) automorphic functions Holomorphic extension of representations: (I) automorphic functions

2002-10-07

arxiv.org/abs/math/0210111v2

Ann. of Math. (2), Vol. 159 (2004), no. 2, 641--724

Mathematics

Representation Theory

84 pages, published version

Scientific paper

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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