The conjectural character formula for discrete-series characters of $\mathrm{SL}_\ell(F)$

Let $F$ be the nonarchimedean local field and let $G=\mathrm{SL}_\ell(F)$, with $\ell$ an odd prime. Let $T$ be an unramified torus, let $\psi\in\widehat{T(F)}$, and let $\gamma\in T(F)$ be regular. Write $r$ for the depth parameter of the representation associated with $\psi$, let $\gamma_{<r}$ denote the corresponding shallow component, let $X_{\psi,<d(\gamma)}^\ast$ be the associated truncated dual datum, let $D_G$ denote the discriminant, and let $\Gamma_{E/F}$ be the Galois group associated with the splitting field $E$ of $T$.

The conjectural character formula. The character value is given by

$$\Theta_\psi(\gamma)=|D_G(\gamma_{<r})|^{-\frac{1}{2}}|D_G(X_{\psi,<d(\gamma)}^\ast)|^{\frac{1}{2}}\epsilon_\psi(\gamma)\sum_{\sigma\in\Gamma_{E/F}}\psi(\gamma^\sigma),$$

where $\epsilon_\psi(\gamma)$ is a root of unity depending on $\psi$ and $\gamma$.

This formula is intended to provide the character values on the unramified torus, where existing character formulas involving orbital integrals are difficult to compute. The paper indicates that the formula is conjectural because current techniques do not fully determine these values; in particular, a direct purely local proof of a key identity used in the torus character table remains to be found.