Subconvexity for sup-norms of automorphic forms on PGL(n)

As it was proved by Sarnak, the supnorm of eigenfunctions of the Laplacian on a compact symmetric
Riemannian manifold can be estimated from above by an appropriate power (given in terms of some
invariants of the space) of their Laplace eigenvalue. Examples show that Sarnak's exponent is sharp
in some cases. However, when the space has also arithmetic symmetries (i.e. Hecke operators) and
we restrict to joint eigenfunctions of the Laplacian and the Hecke operators, one might expect a
better exponent. We prove that a better exponent exists for automorphic forms on PGL(n,R).
Joint result with Valentin Blomer.