Sup-norm bounds for Siegel-Maass forms

Given a Riemannian locally symmetric space, bounds for
eigenfunctions of the Laplace operator or for joint eigenfunctions of
the whole algebra of isometry-invariant differential operators are of
great interest in several areas. For example, sup-norm estimates are
intimately related to the multiplicity problem and to questions of
quantum unique ergodicity. Methods from analysis allow to provide bounds
(nowadays called `generic') which are sharp for certain spaces.

If the Riemannian locally symmetric space is arithmetic and one
restricts the consideration to the joint eigenfunctions of the algebra
of differential operators and of the Hecke algebra then it is reasonable
to expect that the generic bounds can be improved.

The archetypical result of such kind is due to Iwaniec and Sarnak in the
situation of the modular surface and several other arithmetic Riemannian
hyperbolic surfaces, dating back to 1957. In the following years, their
way of approach was (and still is) used to deduce many similar results
for various spaces of rank at most 1, and was only recently adapted to
some higher rank spaces.

The first example of a subconvexity bound for a higher rank setup was
provided by our joint work with Valentin Blomer, which we will discuss
in this talk.