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Estimates of automorphic forms and representation theory


Speaker: Joseph Bernstein, Tel Aviv University and MSRI
Location: 693 Kerr
Start time: Mon, Nov 19 2001, 4:10PM

Let $H$ be the upper half plane with the hyperbolic metric of constant curvature $-1$. We consider the natural action of the group $G = \SLR$ on $H$. \sms Fix a lattice (discrete subgroup) $\Gam\subset G$ and consider the Riemann surface $Y=\Gam\setminus H$. \sms The Laplace-Beltrami operator $\Del$ acts in the space of functions on Y. When $Y$ is compact it has discrete spectrum; we denote by $\mu_1 \leq \mu_2 \leq ...$ its eigenvalues on $Y$ and by $\phi_i$ the corresponding eigenfunctions. (We assume that $||\phi_i ||_{L^2} = 1$.) These functions $\phi_i$ are usually called {\it automorphic functions} or {\it Maass forms} . \sms The study of automorphic functions and the corresponding eigenvalues is important in many areas of representation theory, number theory and geometry. \ms In my talk I will discuss several problems of estimating numbers arising from modular forms. One of these problems is to find good estimates of the triple triple products of automorphic functions. \sms More precisely, let us fix one automorphic function, $\phi$, and consider the function $\phi^2$ on $Y$. Since $\phi^2$ is not an eigenfunction, it is {\it not} an automorphic function. \sms Since $\phi^2\in L^2(Y)$, we may consider its spectral decomposition in the basis $\{\phi_i\}$: $ \phi^2=\sum c_i\phi_i. $ Here the coefficients are given by the triple product integrals: $c_i=\langle\phi^2, \phi_i\rangle=\int_X \phi \cdot \phi \cdot {\overline \phi_i} dx$. \sms I will explain how one can get an estimate of these triple products using representation theory of the group $\SLR$. I will also explain why these triple products are of interest and how they are related to the theory of $L$-functions. \ms The main idea of the method which I will try to explain is to associate to every automorphic function $\phi$ an automorphic representation of the group $G =\SLR$ and study properties of this representation.