Number Theory Seminar

Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}$ tends to $0$ when $\pi$ runs over equivalence classes of irreducible supercuspidal representations of $G(F)$ whose central character is unitary and the formal degree of $\pi$ tends to infinity. In fact something stronger holds under some additional conditions. I give the sketch of the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open compact modulo center subgroup.