Let $\mathbf{ G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. In this paper we study the asymptotic behaviour of the trace characters $\theta _{ \pi}$ evaluated at a regular element $\gamma $ of $\mathbf{ G}(F)$ as $\pi$ varies among supercuspidal representations of $\mathbf{ G}(F)$. Kim, Shin and Templier conjectured that $\frac{ \theta_{ \pi}(\gamma)}{ { \rm deg}(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $\textbf{ G}(F)$ with unitary central character and the formal degree of $\pi$ tends to infinity. For $\textbf{ G}$ semisimple we prove that the trace character is uniformly bounded on $\gamma$ under the assumption, which is expected to hold true for every $\textbf{ G} (F)$, that all irreducible supercuspidal representations of $\textbf{ G}(F)$ are compactly induced from an open compact modulo center subgroup. Moreover, we give an explicit upper bound in the case of $\gamma $ ellitpic.
Categories: math.RT math.NT