$L^4$-norms of eigenfunctions of the Laplacian on hyperbolic Riemann surfaces

Let Y be a compact Riemann surface with curvature -1 and the associated Laplace-Beltrami operator D. Let $f_i$ be an orthonormal basis in $L^2(Y)$ consisting of eigenfunctions of D with the corresponding eigenvalues $m_i$. We prove that the $L^4$-norm of $f_i$ is bounded by a constant independent of the eigenvalue $m_i$. We discuss some application of this result to the spectrum of D. The proof is based on ideas from representation theory of the group $SL(2,R)$ and we plan to explain this connection from scratch. The result is a joint work in progress with J. Bernstein.