TAILIEUCHUNG - Đề tài " Holomorphic extensions of representations: (I) automorphic functions "

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. . | Annals of Mathematics Holomorphic extensions of representations I automorphic functions By Bernhard Kr otz. and Robert J. Stanton Annals of Mathematics 159 2004 641 724 Holomorphic extensions of representations I automorphic functions By Bernhard Krotz and Robert J. Stanton Abstract Let G be a connected real semisimple Lie group contained in its complex-ification 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 Ly- 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 . of triple products of Maafi forms. Introduction Complex analysis played an important role in the classical development of the theory of Fourier series. However even for Sl 2 R contained in Sl 2 C complex analysis on Sl 2 C has had little impact on the harmonic analysis of Sl 2 R . As the K-finite matrix coefficients of an irreducible unitary representation of Sl 2 R can be identified with classical special functions such as hypergeometric functions one knows they have holomorphic extensions to some domain. So for any infinite dimensional irreducible unitary representation of Sl 2 R one can expect at most some proper subdomain of Sl 2 C to occur. It is less clear that there is a universal domain in Sl 2 C to which the action of G on K-finite vectors of every irreducible unitary representation has holomorphic extension. One goal of this paper is to construct such a domain for a real connected semisimple Lie group G contained in its complexification GC . It is .

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