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**Holomorphic extensions of representations. I: Automorphic functions.**
*(English)*
Zbl 1053.22009

One goal of this paper is to construct a universal domain for a real connected semisimple Lie group \(G\) contained in its complexification, to which the action of \(G\) on \(K\)-finite vectors of every irreducible unitary representation has holomorphic extension. The main goal of this paper is to use the holomorphic extension of \(K\)-finite vectors and their matrix coefficients to obtain estimates involving automorphic functions.

The authors extend the technique of Bernstein and Reznikov, which introduced the notion of a maximal invariant seminorm associated to Sobolev norms of vectors in representations. First, by using a more representation theoretic viewpoint, they are able to treat the case of real rank one groups. Secondly, they obtain estimates for the decay rate of Fourier coefficients of Rankin Selberg product of Maass forms for \(G=SL(n, \mathbb R)\) and give a conceptually simple proof of results of Good on the growth rate of Fourier coefficients of Rankin Selberg products on \(SL(2, \mathbb R)\).

The authors extend the technique of Bernstein and Reznikov, which introduced the notion of a maximal invariant seminorm associated to Sobolev norms of vectors in representations. First, by using a more representation theoretic viewpoint, they are able to treat the case of real rank one groups. Secondly, they obtain estimates for the decay rate of Fourier coefficients of Rankin Selberg product of Maass forms for \(G=SL(n, \mathbb R)\) and give a conceptually simple proof of results of Good on the growth rate of Fourier coefficients of Rankin Selberg products on \(SL(2, \mathbb R)\).

Reviewer: Gabriela P. Ovando (Córdoba)