Fourier expansion along geodesics on Riemann surfaces (Q2440583)

For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved. In more detail, for an automorphic function, the invariance under parabolic elements is used to give the standard Fourier expansion, the coefficients of which define the \(L\)-function of the form. This paper considers instead the Fourier-expansion along a hyperbolic element. In other terms, let \(Y\) be a hyperbolic Riemann surface and let \(c\) be a closed geodesic in \(Y\). Consider the Fourier coefficients \(c_k(f) = \int_0^1 f(c(\ell_ct))e^{2\pi ikt}dt\) of a smooth function \(f \in C^\infty(Y)\). Here \(\ell_c\) is the length of the geodesic \(c\). Under the assumption that \(f\) is an eigenfunction of the Laplace operator on \(Y\) with eigenvalue \(\alpha\), one can relate \(c_k\) to an intertwining integral \(I_k^\alpha(f)\), which depends on \(\alpha\) and \(f\), but not on \(c\). There is an automorphic coefficient \(a_k\in\mathbb{C}\), such that \(c_k=a_kI_k^\alpha\). The present paper contains the following three main results: (1) A classification of all intertwining functionals on the dual of the group PGL(2,\(\mathbb{R}\)). (2) A growth estimate \(a_k=O(|k|^{1/2})\) as \(|k|\to\infty\). Here the proof uses the technique of analytic continuation developed by \textit{J. Bernstein} and \textit{A. Reznikov} [Ann. Math. (2) 150, No. 1, 329--352 (1999; Zbl 0934.11023)]. (3) A summation formula which involves the coefficients \(a_k\) and the spectral decomposition in the compact case. Here the proof relies on the uniqueness of invariant trilinear forms as in [\textit{J. Bernstein} and \textit{A. Reznikov}, Mosc. Math. J. 4, No. 1, 19--37 (2004; Zbl 1081.11037)].