Large values of cusp forms on GL(n)

ABSTRACT:

The study of sup norms of eigenfunctions of the Laplacian on compact
manifolds has a long history, the first results dating back to the
60's and the work of Hormander. When the compact manifold is a
negatively curved arithmetic locally symmetric space, sup norms of
eigenfunctions have attracted the attention of number theorists, not
least for their relation to L-functions. We shall be interested in the
size of cusp forms on certain non-compact spaces, namely, congruence
quotients of SL(n,R)/SO(n). These eigenfunctions oscillate on a
sizable bulk of the space and decay rapidly in the cusps. In
transitioning between these two regimes, the oscillation slows and the
form gets large. When n=2, Iwaniec and Sarnak quantified this behavior
for Maass cusp forms, showing, in particular, that their sup norm
grows as a power of the eigenvalue. In work with N. Templier,
we investigate the size of cusp forms in the transition range in
higher rank. Among other results, we obtain lower bounds on the sup
norms of surprising strength for general n.