Uniform bounds on the Harish-Chandra characters

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 $\frac{{\theta }_{\pi }(\gamma )}{\mathrm{d}\mathrm{e}\mathrm{g}(\pi )}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $\mathbf{\text{G}}(F)$ with unitary central character and the formal degree of $\pi$ tends to infinity. For $\mathbf{\text{G}}$ semisimple we prove that the trace character is uniformly bounded on $\gamma$ under the assumption, which is expected to hold true for every $\mathbf{\text{G}}(F)$, that all irreducible supercuspidal representations of $\mathbf{\text{G}}(F)$ are compactly induced from an open compact modulo center subgroup. Moreover, we give an explicit upper bound in the case of $\gamma$ ellitpic.