Analytic Continuation of Representations and Estimates of Automorphic Forms

Abstract

0.1. Analytic vectors and their analytic continuation. Let G be a Lie group and (�,G,V ) a continuous representation of G in a topological vector space V. A vector v ∈ V is called analytic if the functionv : g 7→ �(g)v is a real analytic function on G with values in V. This means that there exists a neighborhood U of G in its complexification GC such thatv extends to a holomorphic function on U. In other words, for each element g ∈ U we can unambiguously define the vector �(g)v asv(g), i.e., we can extend the action of G to a somewhat larger set. In this paper we will show that the possibility of such an extension sometimes allows one to prove some highly nontrivial estimates. Unless otherwise stated, G = SL(2, R), so GC = SL(2, C). We consider a typical representation ofG, i.e., a representation of the principal series. Namely, fix� ∈ C and consider the space Dof smooth homogeneous functions of degree � − 1 on R 2 0, i.e., D� = {� ∈ C ∞ (R 2 0) : �(ax,ay) = |a| �−1 �(x,y)}; we