Holomorphic extensions of representations: (I) automorphic functions

• 成果类型: Article
• 署名作者: Krötz, B; Stanton, RJ
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2004
• 页码: 641-724
• 关键词: FORMS

摘要

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-infinity 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 Maabeta forms.