Analytic continuation of representations and estimates of automorphic forms

0.1. Analytic vectors and their analytic continuation. Let G be a Lie group and (π,G, V ) a continuous representation of G in a topological vector space V . A vector v ∈ V is called analytic if the function ξv : g 7→ π(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 that ξv extends to a holomorphic function on U . In other words, for each element g ∈ U we can unambiguously define the vector π(g)v as ξv(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 λ ∈ C and consider the space Dλ of smooth homogeneous functions of degree λ − 1 on R2 \ 0, i.e., Dλ = {φ ∈ C∞(R2 \ 0) : φ(ax, ay) = |a|λ−1φ(x, y)}; we denote by (πλ, G,Dλ) the natural representation of G in the space Dλ. Restriction to S1 gives an isomorphism Dλ ≃ C∞ even(S), and for basis vectors of Dλ one can take the vectors ek = exp(2ikθ). If λ = it, then (πλ,Dλ) is a unitary representation of G with the invariant norm ||φ||2 = 1 2π ∫ S |φ|2dθ. Consider the vector v = e0 ∈ Dλ. We claim that v is an analytic vector and we want to exhibit a large set of elements g ∈ GC for which the expression π(g)v makes sense. The vector v is represented by the function (x2 + y2) λ−1 2 ∈ Dλ. For any a > 0 consider the diagonal matrix ga = diag(a −1, a). Then