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Estimates of automorphic functions. (English) Zbl 1081.11037
Consider \(\text{SL}(2,\mathbb{R})\) acting on the upper half plane \(\mathbb{H}\) by fractional linear transformations and let \(\Gamma\) be a discrete cocompact subgroup. We set \(Y=\Gamma\setminus\mathbb{H}\). The Laplace operator acts on functions on \(Y\) with discrete eigenvalues \(0< \mu_1\leq\mu_2,\dots\). Let \(\phi_i\) denote the normalized eigenfunction with eigenvalue \(\mu_i\). Fix two eigenfunctions \(\phi\) and \(\phi'\) and consider \[ c_i= \int_Y \phi\phi'\phi_i \,dy. \] Define \(\lambda_i\) such that \(\mu_i={1\over 4}(1- \lambda^2_i)\), and define \(b_i= |c_i|^2\exp({\pi\over 2} |\lambda_i|)\). The main theorem of this paper states that there exists a constant \(A\) such that for any \(T> 0\), \[ \sum_{T\leq |\lambda_i|\leq 2T} b_i\leq A. \] The authors claim that this theorem is a stronger version of the main theorem in [\((*)\) J. Bernstein and A. Reznikov, Analytic continuation of representations and estimaes of automorphic forms, Ann. Math. (2) 150, No. 1, 329–352 (1999; Zbl 0934.11023)] where they prove that \(\sum_{|\lambda_i|\leq T} b_i\leq C(\ln T)^3\). An important step of the proof is to interpret \(c_i\) as the value of the \(\text{PGL}(2,\mathbb{R})\)-invariant linear functional on the tensor product three principal series representations of \(\text{PGL}(2,\mathbb{R})\), evaluated at the tensor product of the spherical vectors. Such linear functionals are known to be one-dimensional.
Estimates of the \(c_i\)’s have important number-theoretic implications. This is explained in the introduction of this paper as well as in [\((*)\)].

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
11F12 Automorphic forms, one variable
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