KST character-decay conjecture for supercuspidal representations

Let $G$ be a $p$-adic group with compact center $Z$, let ${\rm Irr}^{\rm sc}(G)$ denote its supercuspidal irreducible representations, let $\deg(\pi)$ be the formal degree, and let $G_{\mathrm{rs}}$ and $G_{\mathrm{ell}}$ denote the regular semisimple and elliptic elements of $G$, respectively. Write $\Theta_\pi$ for the character of $\pi$ and $D(\gamma)$ for the discriminant.

KST character-decay conjecture. The following assertions hold:

1. For every $\gamma\in G_{\mathrm{rs}}$,

$$\lim_{\substack{\pi\in {\rm Irr}^{\rm sc}(G)\\ \deg(\pi)\to\infty}}\frac{\Theta_\pi(\gamma)}{\deg(\pi)}=0.$$

Equivalently, for every $\epsilon>0$ there exists $d_\epsilon>0$ such that $|\Theta_\pi(\gamma)/\deg(\pi)|<\epsilon$ whenever $\pi\in {\rm Irr}^{\rm sc}(G)$ and $\deg(\pi)>d_\epsilon$.

2. For every bounded subset $\mathcal B\subset G$, there exist $\nu>0$ depending only on $G$ and $C_\mathcal B>0$ depending only on $G$ and $\mathcal B$ such that

$$|D(\gamma)^{1/2}\Theta_\pi(\gamma)|\leq C_\mathcal B\deg(\pi)^{1-\nu},\qquad \forall\pi\in {\rm Irr}^{\rm sc}(G),\ \forall\gamma\in G_{\mathrm{rs}}\cap\mathcal B.$$

3. There exist $\nu>0$ and $C_{\mathrm{ell}}>0$, depending only on $G$, such that

$$|D(\gamma)^{1/2}\Theta_\pi(\gamma)|\leq C_{\mathrm{ell}}\deg(\pi)^{1-\nu},\qquad \forall\pi\in {\rm Irr}^{\rm sc}(G),\ \forall\gamma\in G_{\mathrm{rs}}\cap G_{\mathrm{ell}}.$$

The conjecture predicts uniform control, with power saving in the formal degree, for characters of supercuspidal representations on regular semisimple elements; the paper proves related bounds for supercuspidal representations constructed by Yu, while the full assertions remain open in the stated generality.