Asymptotic behavior of supercuspidal characters

This is a report on joint work in progress with Julee Kim and Nicolas Templier. The irreducible smooth representations of a p-adic reductive group are said to be supercuspidal if they do not appear in any induced representation from a proper parabolic subgroup. While it is still an open problem to obtain a precise character formula for them (apart from some special cases), I will explain that we can prove a reasonable upper bound and a limit formula as the formal degree tends to infinity, for a large class of supercuspidal representations. As an application we establish an equidistribution result and a low-lying zero statistics for L-functions in a new kind of families of automorphic representations.